KURT MAHLER
Department of Mathematics, Institute of Advanced Studies
Australian National University
p-adic numbers and
their functions
Second edition
CAMBRIDGE UNIVERSITY PRESS
CAMBRIDGE
LONDON NEW YORK NEW ROCHELLE
MELBOURNE SYDNEY

Ifct^^jy.^.^^-^-^^
UNIVERSITATS-
BiBLIOTHBK
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,--6261
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Cambridge University Press 1981
First published 1973
Second edition 1981
Photosetting by Thomson Press (India) Limited, New Delhi
Printed in Great Britain at the University Press, Cambridge
Library of Congress Cataloguing in Publication Data
Mahler, Kurt.
p-adic numbers and their functions.
(Cambridge tracts in mathematics; 76)
First published in 1973 under title: Introduction
to p-adic numbers and their functions.
Includes index.
1. p-adic numbers. 2. Numerical functions.
I. Title. II. Series.
QA241.M22 1980 512'^f 79-20103
ISBN 0 521 23102 7 ^d^dition
(ISBN 0 521 20001 6/1 st edition)
i i
^^^b^*-^*****'*
Standort: D 31
Signatur: TFC 1712 (2)
Akz.-Nr.: 84/9753
ld"Nr-: C8 20315128

Sayings of Confucius from the Analects (Lun-Yii)
The Master said: 'The ancients were reserved in their speech lest their
actions might not come up to their words.' (IV-21)
The Master said: 'A cornered vessel that has no corners. What a
cornered vessel! What a cornered vessel!' (VI-23)
The Master said: 'It is not easy to find a man who after three years of
study has never thought of reward.' (VIII-12)
The Master said: 'In hearing lawsuits, I am no better than other men,
but my aim is to bring about the end of lawsuits.' (XII-13)
'What would you say of those now in government?' 'Ugh', said the
Master, 'Those ricebags! They are not worth taking into account.' (XIII-
20)
The Master said: 'In education there is no class distinction.' (XV-38)
Taken from The sacred books of Confucius, by Ch'u Chai and Winberg
Chai, New York, 1965

Contents
page
Preface ix
Preface to the first edition xi
PART I: NUMBERS 1
1 g-adic values of rational numbers 3
2 Pseudo-valuations and valuations 17
3 g-adic and /?-adic numbers 33
4 The arithmetic of Qg and Qp 42
5 The decomposition of Qg into /?-adic fields 53
6 The quadratic extension fields of Qp 65
PART II: FUNCTIONS 83
7 Elementary topological properties of R, Qp, and Qg 85
8 First properties of continuous g-adic functions 104
9 The interpolation series of a g-adic function 121
10 Characterisation of functions with the properties
(N) and (W) 138
11 Further properties of /?-adic functions 160
12 Remarks on functions of two variables 180
13 The derivative of a function on J or I 188
14 Higher derivatives 220
15 The Dieudonne integral 234
16 The van der Put series and integral 252
17 Definite integrals and difference equations 275
18 Functions on the quadratic extension fields of Qp 301
References 316
Notation 318
Index 319

Preface
The main changes in this new edition concern the second part on
functions of a p-adic variable where new chapters with new results
have been added.
The first five chapters have been checked for errors, but have
otherwise been little changed, except that more examples have been
added. In a new Chapter 6 all the distinct quadratic extension fields
of the p-adic field have been determined, and the p-adic valuation has
been continued to these extension fields.
The second part, on functions, is, however, almost entirely new. It
starts with some basic facts on metric spaces and ordered rings, for the
purpose of making the similarities and the differences between real
and p-adic functions more intelligible. For the same reason I deal in
Chapters 8-10 with the more general case of functions of a gr-adic
variable. In Chapter 10, necessary and sufficient conditions for such
functions to be uniformly approximable by polynomials are obtained.
It turns out that every continuous p-adic function, but not every
continuous gr-adic function, has this property. For this reason, the
remaining chapters are restricted to functions of a p-adic variable.
Chapters 11-14 treat first and higher derivatives. Then, in Chapters
15 and 16, two essentially different definitions of the indefinite
integral, due to Dieudonne and van der Put, respectively, are studied
and the surprising fact is proved that both lead to the same result.
Finally Chapter 17 is concerned with a special kind of definite integral
and with its relation to the solutions of a certain difference equation.
The last chapter deals with functions on the integers of a quadratic
extension of the p-adic field which can be expanded into interpolation
series.
My intention has been to write an elementary and self-contained
text on the calculus of p-adic functions. The main stress lies on
continuous and differentiable functions, while little is said about

x Preface
analytic functions because there are now many excellent books on
these. The book should present little difficulty to students who have
taken courses on elementary calculus and algebra, say to second year
honours students at British universities.
In writing this book, I have been greatly indebted to R. Bojanic for
his help with the first ten chapters. Unfortunately this cooperation
could not be continued owing to the distance between Columbus and
Canberra.
My grateful thanks are also due to other colleagues and in
particular to J. W. S. Cassels, C. S. Weisman, and M. Waldschmidt
for many helpful remarks, and to Arnold E. Ross for his help with
checking the manuscript. I also wish to express my gratitude to Miss
Sutherland of C.U.P. for her careful preparation of the manuscript
for the printer.
Canberra K. Mahler
November 1978

Preface to the first edition
This set of notes contains an elementary introduction to the theory of
p-adic numbers and their analysis. These numbers were introduced by
K. Hensel some eighty years ago and have slowly become of
importance in more and more parts of mathematics. Nevertheless,
while many recent books on algebra have short chapters or
paragraphs on the subject, a really good introduction to p-adic numbers
from the standpoint of elementary analysis still does not seem to exist.
Hensel's book, Zahlentheorie (1913) is still one of the best
elementary books, but it has become somewhat out of date. An excellent
introductory presentation from the standpoint of valuation theory is
given in G. Bachmann's book, Introduction to p-adic Numbers and
Valuation Theory (1964), and another detailed treatment from this
same viewpoint can be found in the first chapters of my little book on
Diophantine Approximations (1961). For a more advanced
presentation, the neatest approach to the p-adic field and its algebraic
extensions can perhaps be found in the first chapter of O'Meara's
book, Introduction to Quadratic Forms (1963). Of special interest is the
new book by A. F. Monna, Analyse Non-archimedienne (1970).
We shall begin by studying the g-adic rings and p-adic fields, and
then finally investigate continuous and differentiable functions of a
p-adic variable.
A course similar to this presentation was given repeatedly at the
Ohio State University.
Columbus, Ohio K. Mahler
November 1971

I. NUMBERS

g-adic values of rational numbers
(i)
1 Rational integers
We shall try to develop everything from first principles, but there are a
few ideas from algebra and later from elementary analysis which will
be assumed.
Z denotes the ring of rational integers
-2-10123
That Z is a ring (more exactly a commutative ring with unit element 1)
means that if a, b, c lie in Z, then
a + b, a — b, ab lie in Z;
0 + a = a, a — a = 0, la = a;
a + b = b + a9 ab = ba;
(a + b) + c = a + (b + c), (ab)c = a(bc);
a(b + c) = ab + ac9/ (a + b)c = ac + bc.
(The third, fourth, and fifth lines state the commutative, the
associative, and the distributive laws of addition and multiplication.)
A product of integers vanishes if and only if at least one factor
vanishes; i.e. there are no zero divisors in Z.
In general, if a =£ 0 and b are integers, then b/a = c need not be an
integer. If c is in fact an integer, then we say that a divides b and write
a\b,
and we otherwise say that a does not divide b and write
ajb.
Any two or more integers al9a2,... ,an not all zero have a greatest
common divisor d =^(au a2,..., an) > 0 such that
d\ak for k = 1, 2,..., n;
but d' does not divide all of if d' > d.

4 g-adic values of rational numbers
An important theorem states that there are n integers xl9 x2,..., xn
such that
axxy + a1x1 + ...+ anxn = d.
From this property one can easily deduce that
if a|be and (a, ft)=l, then a\c.
Of particular importance are the primes which are integers p > 1
which have + 1 and + p as their only divisors. The first primes are
2,3,5,7,11,13,17,19,23,29,...,
and there are infinitely many. An integer > 4 which is not a prime is
said to be composite; e.g. 4 = 2x2 and 6 = 2x3 are composite. Every
composite number is the product of finitely many primes, and this
representation is unique except for the order of the factors.
On adjoining to Z the set of all rational numbers a/b where a and
b > 1 are integers, we obtain the field Q of all rational numbers. In this
field, addition, subtraction, multiplication, and division are always
possible in just one way, except that division by 0 is not allowed.
The rules (1) hold also in Q. Every number in Q has a unique
simplified representation
a/b, where a, b are in Z, b > 1, (a, ft) = 1.
2 Real numbers
The field Q of rational numbers can next be extended to the field R of
real numbers. Here real numbers are limits of convergent sequences of
rational numbers; e.g.
i 111 r £ 1
e = 1+T7 + 2T + 3T + -=hm ^fcT
is a real number, and so are all rational numbers and numbers like
^/2,7i, e71, and infinitely many others. The rules (1) hold also in the
field R.
The elements of R (and similarly those of Q) split into three disjoint
sets consisting, respectively, of
the one number 0,
the positive numbers > 0, and
the negative numbers < 0.

Two lemmas 5
If a is positive, then — a is negative, and vice versa. If further a and b
are positive, so are a + b and ab.
Every positive number a can be written as a decimal fraction
00
a= £ ak10~k,
k = f
where f is a certain integer (= 0, > 0, or < 0), and the coefficients or
digits ak have one of the ten values
0, 1, 2, 3, 4, 5, 6, 7, 8, or 9.
This representation can be generalised to the representation to the
basis g. Here (as throughout this chapter) g is an integer > 2, and the
positive number a is written in the form
00
a= Z ak9~k-
k=f
Here / is again a certain integer which depends on a, and the
coefficients ak are now restricted to the set {0,1,2,...,0-1}.
Elements of this set will be called digits to the basis g, or simply digits.
We note that the exponents — k of g in this formula for a are
descending and tend to — oo. In the remainder of this chapter we shall
be concerned with similar formulae for rational numbers (not for
general real numbers) where the exponents of g are ascending and tend
to + 00.
The simplest case of such a representation is that of a positive
integer
n
<*= Z akGk>
k = 0
where n is a certain non-negative integer, and the ak are again digits.
The more general formula
n
k= -/
where f is a positive integer, represents a rational number with the
denominator gf.
3 Two lemmas
Lemma 1 Let G, r, s fee integers such that G > 2, s > 1, and

6 g-adic values of rational numbers
(G, s) = 1. Then a unique pair of integers A and R exists such that
0<A<G-1, r/s = A + (GR/s). (2)
Proof of the existence of A and R. Since (G, s) = 1, there are integers x
and y such that
sx + Gy = 1, hence s(rx) + G(ry) = r,
and therefore also
s(rx — kG) + G(ry + ks) = r. (3)
Choose the integer k such that
0 < rx - kG < G - 1,
and with this choice of x, y, and k put
A = rx — kG, R — ry + ks.
Then, by (3),
sA + GR = r, r/s = A + (G R/s),
as asserted.
Proof of the uniqueness of A and R. If a second pair of integers A' and
R' exists such that
0<A'<G-1, r/s = A' + (GR'/s\
then
s(A' -A) = G(R - R').
Since (G, s) = 1, this implies that
GM'-A
On the other hand, 0 < A < G - 1 and 0 < A' < G - 1, hence
-(G-1)<^'-^<G-1.
It follows then that A' = A and hence also R' = R, as was to be proved.
Lemma 2 Let r, s9 and g be integers such that
r^O, s>l, (r,s)=l, g>2.
Then there exists a unique integer f (= 0, > 0, or < 0) and a pair of
integers R9 S such that
gfr/s = R/S9 g\R9 (R,S) = to,S)=l.
Proof. lig\rr denote by g4*, where </>>!, the highest integral power of

Two lemmas 1
g that divides r. On putting R = g~<f)r and S = s, we have gj[R and
(#, 5) = (g9 S) = 1, and further
g-*r/s = R/S,
proving the assertion with f — — </> < 0.
Next, if g\r and (g, s) = 1, the assertion holds with f = 0, R = r,
Finally let grJV and (g, s) > 1. Then s can be factorised as
5 = 5^2 (sx >0, s2>0)
such that all prime factors of sx divide g, but that (g, s2) = 1. Similarly
0 can be written as a product
0=0102 (01>O,02>O)
such that all prime factors of gl divide sl9 but that (g2, sx) = (#2, s) = 1.
There is a smallest positive integer \jj such that
s^g* and hence also s±\g%.
In the equation
0V = (0tAl)(027S2)r
the quotient g\/sl is an integer. Put therefore
R = (0iMgir and S = s2.
Then
g*r/s = R/S, glR, (R,S) = (g9S)=l9
proving the assertion with f = \jj > 0.
Definition If the notation is as in Lemma 2, and if a = r/s9 so that
a / 0, then we put
\a\„ = qf, 101=0.
\J 7 I IU
10 » ? I~|0
The so-defined function \x\g of x is called the g-adic value of x.
From the proof of Lemma 2, the inequality
\r/s\g<l
holds exactly when
(g, s) = 1.
Rational numbers with this property are called g-adic integers.

8 g-adic values of rational numbers
4 Properties of the g-adic value
The absolute value \a\ of a rational (or real) number a is defined by
|a|=a if a>0; \a\=—a if a < 0.
It has the following properties.
(A) |0| = 0; |a|#0 if a#0.
(B) \a ± b\ < \a\ + \b\ (triangle inequality).
(C) \ab\ = \a\\b\.
(D) |na| = n|a| for n = 0, 1, 2,...
In the terminology of the next chapter, \a\ is an archimedean valuation.
As we show now, the #-adic value \a\ of a rational number satisfies
somewhat analogous laws.
The first three properties are easy consequences of the definition
and of Lemma 2. It is obvious that the analogue of (A) is satisfied.
(a) \a\g = 1 if and only if a = r/s where gfr, (g, s) = 1.
(b) \a\ = gf if and only if \gfa\g = 1.
(c) For every integer </>,
\g*a\g = g~*\a\g.
Proof. The assertion is obvious when a = 0. Excluding this case, let
\a\n = gf. Then
i \g o
\gfa\g = \gf~*-0H = 1 whence l?H = ^/"0 = flf"0N«-
These three properties have no analogues for the absolute value.
(d) \a±b\g< max (\a\g9 \b\g) (triangle inequality).
Proof. When a = 0 or b = 0, this inequality obviously holds because
| — b\ = \b\g. So assume that a / 0 and b / 0 have the #-adic values
\a\g = gf and \b\g = g<f>
where, without loss of generality, /<</>. The denominators of both
g^a and g^b are prime to g; hence the same is true for the denominator
ofg+ia + b). But then
\gHa±b)\g<i9
and it follows from (c) that
\a±b\g<g<f) = max(\a\g9\b\g),
as asserted.

Properties of the g-adic value 9
If \a\g^ \b\g9 say \a\g < \b\g9 that is/ < 0, more can be proved. Now
the numerator of g^a, but not the numerator of g^b, is divisible by g.
Hence neither is the numerator of g* (a ± b) a multiple of g9 and it
follows that
\a±b\g = \b\g = max (\a\g,\b\g) if \a\g<\b\g.
The triangle inequality for \a\g is thus stronger than that for \a\9 for (d)
implies of course that
\a±b\g<\a\g + \b\g.
By induction on the number of terms, the triangle inequality (d) can
easily be generalised to sums of more than two rational numbers, in
the form
|ax +a2 + ... +an\g < max (1^1^, \a2\g9..., \an\g).
On putting here all ak equal to the same rational number a9 it follows
that
\na\g < \a\g (n= 1, 2, 3,...).
This inequality is quite different from the property (D) of the absolute
value.
The next two properties deal with the #-adic value of products.
(e) \ab\g<\a\g\b\g.
Proof The assertion is again obvious if a = 0 or b = 0. Let then a / 0
and ft # 0, say
K = gf, \b\g = g*.
Then
gfa = R/S and g(f>b = P/I
where
gIR, (g,S)=l9 and g\P9 (g9Z)=l9
hence also
Therefore (e) follows from
gf +fab = RP/SZ.
(f) If g = p is a prime, then \ab\p = \a\p\b\p.
Proof The properties pfR and pfP imply now that also pJRP.

10 g-adic values of rational numbers
The product equation (f) does not in general hold when g is not a
prime. By way of example,
2|4=1, but |2x2|4 = i<|2|4x|2|4=l,
2|6 = |3|6 = 1, but |2x3|6=i<|2|6x|3|6 = l.
In the terminology of the next chapter, \a\ and \a\ are a non-
archimedean pseudo-valuation, and a non-archimedean valuation,
respectively.
5 The g-adic expansion of a rational number
We begin with the case when a = r/s is a rational #-adic integer, thus
when
(9, s) = 1.
Denote by n an arbitrary positive integer. On applying Lemma 1 with
G = gn, it follows that there exists a unique pair of integers An9 rn such
that
a = An + gnrn/s, 0<An<gn-l,
hence by \rn/s\g < 1 also
\a-An\g<g~n.
In the same way, there exists a unique pair of integers An + 1, rn + 1 such
that
a = AH+1+gn+1rH+1/s9 0<An+1<gn+i-l,
and hence also
\a-AH + 1\g<Zg-i"+1>.
From these formulae,
-gn<0-(gn-l)<An+1-An<(gn + 1-l)-0<gn+\
and
14,+ 1- An\g = \(a ~ An) ~ (a ~ An + l)\g
<mzx(\a-An\g9 \a-An+l\g)<g-".
Hence the integer An+1— Anis divisible by gn9 say
An+l=An+ "nQ\ (4)
and here
-gn<angn<gn+1.

The g-adic expansion of a rational number 11
Therefore
0<an<g- 1,
so that an is a digit 0, 1, 2,..., g — 1.
On applying the formula (4) repeatedly, with A0 = 0, a0 = Al9 we
find that
A1=aQ9 A2=a0 -\- a^g, A^=a0-\-a1g-\-a2g >•••
and generally
^n = a0 +a±g + .. .-\-an_1gn~1 (n = 1, 2, 3,...),
where the afc are digits. This representation is unique, and from it,
a = r/s = a0 + ayg + ... + an _ ^gn ~ l + #n(rn/s) (rc = 1,2, 3,.. .), (5)
where all the afc are digits, and rn is an integer.
This representation (5) can be extended to all rational numbers
a = r/s. Assume that a is not a g-adic integer, but has the g-adic value
\a\g = 9f> where f > 0.
Then, by Lemma 2,
gfa = R/S9 where #/#, (#, 5) = (g, S) = 1.
Hence, by (5), R/S has for every positive integer N a representation of
the form
R/S = b0 + big +... + bN _ lQN ~ l + gN(RN/S),
where the bk are digits, and RN is a certain integer.
Choose here N = f + n and put
ak = bk + f (k=-f,-f + l9...9n-l).
It follows then that
a = a_ fg~ f + a_ f + ±g~ f + l + ... + a0 + ayg
+ ...+an_1gn-1+gn(Rf + n/S). (6)
Here the coefficients ak are digits, n can be any positive integer, and
Rf + n is an integer.
We call this representation a g-adic expansion of a. The suffixes of
the terms akgk have been selected such that the terma0 is free of factors
gj where j ^ 0.
The expansion (6) implies that
n- 1
a
~ Z akGk
k= ~f
<g~n- (7)
9

12 g-adic values of rational numbers
Since then
lim
n -*• oo
n- 1
k= -f
= 0,
9
we may formally allow n to tend to infinity and symbolically write
00
or in an abbreviated form,
a — a_ fa_ f + l .. .a0, a1a2a3 ... (g\ (8)
where the comma stands after the digit a0. When a is a g-adic integer,
this abbreviated form simplifies to
a = a0, a1a2a3... (g).
If \a\ < 1, one or more of the first digits a0, aua29... may be 0, and if
a = 0, then all the digits vanish.
A detailed study of. the #-adic expansion will be given later.
6 Periodicity of the £-adic expansion of a rational number
The g-adic expansion (8) is said to be periodic, of period length P,
where P is a positive integer, if there exists a suffix m such that
The ordered sequence of digits
amam +1-' 'am + P - 1 (")
is then called a period of a.
With P also every multiple nP9 where n is a positive integer, is a
period length since we may simply repeat the period (9) n times. For
this reason one chooses P as small as possible. Also, instead of starting
the period at the suffix m, it may be started at any later suffix, leading
possibly to a cyclic permutation of the original period (9).
As a short notation for periodicity, we write (8) in the form
a — ci- fd_ j- + 1 . .. a0, a1 a2 • .. am _ \&m&m + i • • • ^m + p - i \g\ (1^)
where the bar is placed over the period.
Theorem 1 A g-adic expansion (8) represents a rational number if
and only if it is periodic.

Periddicity 13
Proof. First assume that the expansion is periodic, say has the form
(10). On terminating the expansion after n repetitions of the period, we
obtain a number a(n) which is given explicitly by
m — 1 m + nP — 1
a(n)= X akgk+ £ akgk.
k = — f k = m
Here, by periodicity,
m + nP-l m + P-1
X afc/= X ak(gk + gk + p + gk+2P+... + gK + (»-»p),
k = m k = m
where the geometric progression
,nP
gk + gk+P + tf + 2P + m9m+gk + (n-l)P = Lji_k
1-0
Hence
m — 1 m + P — 1 ~fc
«(*)= Z ^ + Z a^—p(i-#nP),
k= -f k = m i ~ 9
whence
m — 1 m + P — 1 ~fc
a(n)- Z <W*- Z akT—Zl
k= - f k = m l ~ (J
<g-m-nP
As n tends to infinity, a(n) becomes the expansion (10), and so we find
that a is the rational number
m — 1 m + P — 1 ~fc
a= Z a*0* + Z a
k= - f k = m 1 — 9
Secondly, we shall prove that the #-adic expansion of a rational
number is periodic. It evidently suffices to consider a rational number
a = r/s which is a #-adic integer so that (r, s) = (g, s) = 1.
By § 5, there exists to every positive integer n a pair of integers An, rn
such that
a = r/s = AH + g"(rJs)9 0<An<gn-l,
hence
and therefore
(r - fa" - l)s)/0" < rn < r/0".

14 g-adic valines of rational numbers
For sufficiently large n this implies that
- s < rn < 0,
so that the integer rn has only finitely many possibilities.
Further, with the digit an, and with n replaced by n + 1,
a = An + g^ = An+1+gn+ir^^ = An+ang- + g"^r^^.
s s s
From this it follows that for all n,
rn=ans + grn+l.
Since rn has only finitely many possible values, there exist a suffix m
and a positive integer P such that
rm~rm + P>
hence
(11)
so that
\am ~am + P)S = Qvm + P + 1 — Vm + l)'
It follows then from (g, s) = 1 that #|am —am + P.
Here both am and am + P are digits 0, 1,..., g — 1, and therefore
^m — ^m + P'
On substituting this in (11), we next find that also
rm+ 1 ~rm + P + 1>
which is the original formula rm = rm + p, but with m replaced by m + 1.
We can thus repeat these considerations successively for m, m + 1,
m + 2,... and find that both
rn = rn + p and an = an + P (n = m, m + 1, m + 2,...),
which proves that not only the sequence of the digits ak9 but also that
of the numerators rk has for«>ma period of length P.
7 Numerical examples
A few numerical examples will explain the last investigations,
First let a = f and g = 10; since r = 2 and s = 7, the conditions #JV

Numerical examples
15
and (0, s) = l are satisfied, and therefore \a\g = 1. We find that
2
7
4
7
6
7
2
7
3
7
1
7
7
4
7
6 + 10(
8 + 10(
2+10(
4+10(
1 + 10(
7 + 10(
5 + 10(
8 + 10(
f)
f)
4)
4)
(12)
etc. The last equation is identical with the equation (12), and it follows
that P = 6 and
f = 6, 82417 5 (10).
Conversely,
6,82417 5 (10)
8 + 2 x 10 + 4 x 102 + 1 x 103 + 7 x 104 + 5 x 105
= 6+10
1-106
5714280 40_2
~ 999999 ~ ~~T~1'
As a second example, let r = — 1, s = 1, and g > 2. For every integer
rc> 1,
(0«-1)/(0-1)= 1+0+ 02+... + 0«-i,
hence
-1=(^-1) + (^-1)^ + (^-1)^ + ... + (^-1)^^+^^(-1),
and therefore P = 1 and
-1 = (0-1),(0-1) (0).
On dividing this equation by g — 1, it further follows that
-1/(0-1) = 1,1 (0).
In both these examples the period could have been started at the digit
before the comma.

16 g-adic values of rational numbers
8 Problems
1 Find the g-adic expansions and periods of the following rational
numbers for the given values of g.
| and ^ = 5; ^- and g = 6; ^- and g = 7; f and g = 2;
|| and 0 = 7; iff^r and 0 = 9.
2 # > 2 is a fixed integer. Find the 0-adic expansions and periods for
1/(02 + 1); g/(g - 1); g/(g2 - l)2; (¾*_- 20 - 1)/(02 - 1).
3 g > 2 is a fixed integer, and a, ft, c are digits 0,1,..., 0 — 1. Find
the 0-adic expansion and period of (a — bg — cg2)/(g2 + 1), and
determine the numbers 0, abc (g) and a, abbcc (g).
4 Determine the numbers 12,123456(7) and 0,0012345679(10).
5 Determine the 0-adic values of the following rational numbers: "
and 0 = 7; fff and 0 = 11; £§$ and 0 = 36; 1¾ and g = 91; i^o"
and 0 = 90.
6 Find |0mL„ and |0" m| n, where m and rc are positive integers, and g
is an integer at least 2.
7 0 > 2 is a fixed integer. For which rational numbers a is
8 ft > 0 is an integer. Prove that
\n\Y\\n\p=\,
p
where the product is extended over all primes.
9 Denote by [x] the integral part of x, i.e. the largest integer not
exceeding x. It is proved in number theory that, for every positive
integer n and every prime p, the highest power pN of p dividing n! is
given by iV = [n/p~\ + [n/p2~\ + [n/p3~] + ... where the series
breaks off after finitely many terms.
For which n is (i) |n!|2 = 2"(n" X); (ii) |n!|5 = 5" 19?
Determine the 6-adic value of 100! and of 1/100!
10 Put sn= Yj ~T' ^ry to Prove tnat \sn\i -*0 as fl-^ 00.
k= 1 ^

2
Pseudo-valuations and valuations
1 Definitions and properties
Let K be a commutative ring with the unit (identity element) 1.
Definition A map w from K into the set of non-negative real
numbers is called a pseudo-valuation if it has the following
properties.
w(0) = 0, w(a)>0 if aeK is not 0. (1)
w(a ± b) < w(a) + w(b) for all a.beK. (2)
w(ab) < w(a)w(b) for all a,beK. (3)
If (2) holds in the stronger form
w(a + h)<max(w(a),w(h)) for all a9beK9 (2')
then w is said to be a non-archimedean pseudo-valuation; otherwise it is
said to be archimedean.
If further (3) holds in the strengthened form as an equation,
w(ab) = w(a)w(b) for all a,beK, (3')
then w is called a valuation.
By (1), w(l) > 0. Therefore by (3) and (3'),
w(l) = w(l x 1) < w(l)w(l) for pseudo-valuations,
w(l) = w(l x 1) = w(l)w(l) for valuations w,
and hence
w(l) > 1 for pseudo-valuations,
w(l) = 1 for valuations.
Next, by (2),
w( — a) = w(0 — a) < w(0) + w(a) = w(a)

18 Pseudo-valuations and valuations
and similarly
w(a)< w( — a),
hence
w( — a) = w(a) for ag^.
In particular, when w is a valuation, then
w(l) = w(-l)=l.
Again by (2), since a = b + (a — b),
w(a) < w(b) + w(a — fe), and similarly w(fo) < w(a) + w(a — fe),
hence
|w(a)- w(b)\ <w(a-b). (4)
When w is a non-archimedean pseudo-valuation or valuation^ the
inequality (2') can be strengthened to
w(a±b) = max(w(a),w(b)) if w(a)^w(b). (2")
For let say w(a) < w(b). Then both
b=a — (a — b)9 hence w(fe) < max (w(a — ft), w(a)),
and
b = (a + b) — a9 hence w(ft) < max (w(a + ft), w(a)),
and therefore each of the two assumptions that w(a ±b)< w(b) leads
to a contradiction.
Each of the properties (2), (2'), (3), and (3') may be applied
repeatedly and leads for every positive integer n and any n elements
al9 a2,... ,an of K to
n
w\ £ ak)<
,k = 1
< max w(ak) if w is non-archimedean,
k
and
t
n
n
w\ I\ak){
vfe= 1
< n wK)>
n
= PJ w(ak) if w is a valuation.
k= 1

Sequences in K 19
2 Sequences in K
We begin now with the study of three important kinds of sequences
{an} = {al9a2,a39...}
of elements an of K not necessarily all distinct. (The suffix n will
occasionally run over other increasing sequences of integers.)
Definition A sequence {an} is said to be
w-bounded if there is a constant C > 0 such that
w(an) < C for all n;
a null sequence if
lim w(an) = 0;
n -*• oo
a fundamental sequence if
lim w(am— an) = 0,
m -*• oo )
«-*• oo
(A null sequence is also said to have the w-limit 0; and a fundamental
sequence is said to be w-convergent.)
By the definition of real limits, this means that {an} is a null
sequence if and only if, given any e > 0, there is a positive number p(e)
depending only on e such that
w(an) < e for all n > p(e);
and {an} is a fundamental sequence if and only if, given any e > 0, there
is a positive number g(e) such that
w(am —an)< e for all m>q(e) and all n > q(e).
It is clear that if the sequence {an} has one of these three properties,
the same is true for the sequence
{ — an) = { — #i, — #2, ~ a3> - • •}-
The three types of sequences satisfy the following laws,
(a) Every fundamental sequence is bounded.
Proof. Let e and g(e) be as in the definition, and let q0 be any integer
greater than g(e). If n > q0,
0 ^ w(an) = w(aqo + (an- aj) < w(aj + w(an - aj ^ w(aqo) + e,
= Cx say.

20 Pseudo-valuations and valuations
Now, for all suffixes n,
0 < w(an) < max (w^), w(a2),..., w{aqo _ x), CY)9 = C say,
as asserted.
(b) Every null sequence is a fundamental sequence.
Proof Let e and p(s) be as in the definition, and assume that both m
and n are greater than /?(f e). Then
0 < w(am - an) < w(a J + w(an) < \z + \e. = e,
as asserted.
(c) Let {an} fee a fundamental sequence, and let {nl9 n2, n3,...} be an
increasing sequence of positive integers. If the subsequence
\ani,an2, an3,...)
of {an} is a null sequence, then {an} itself is a null sequence.
Proof Let e, p(e), and g(e) be as earlier defined; let k be so large a suffix
that nk > max (p(|e), q(^e))9 and let m > q(^e). Then
0 < w(aj = w(ank + (am - anJ) < w(ank) + w(am - aj < \z + ^e = e,
as asserted.
By this theorem a fundamental sequence which is not a null
sequence cannot contain a subsequence {ani, an2, an3,...} which is a
null sequence. Hence the following result holds.
(d) If {an} is a fundamental sequence, but not a null sequence, then
there exist two positive numbers c and N such that
w(an)>c if n>N.
The next two theorems are concerned with operations on
fundamental sequences and null sequences.
(e) If {an} and {bn} are fundamental sequences, then so are
K + K), K~fen}> and {anbn}.
Proof. As m and n tend to infinity,
w(am - an) -► 0, w(bm - bn) -+ 0.
Hence the assertion for sum and difference follows from the formulae
0 < w((am + bm) - (an + bn)) < w(am-an) + w(bm - ftn)->0,
0 < w((am - b J - (an - bn)) < w(am - an) + w(bm - bn) -► 0.
In order to prove the assertion also for products, we note that, by (a),
there exist two positive numbers Cl and C2 such that w(an) < Cl and

The ring {K}w 21
w(bn) < C2 for all n. Therefore
0 < w(ambm - anbn) = w(am(bm - bn) + (am - an)bn) < w(am)w(bm - bn)
+ w(am -an)w(bn) < w(fem - bn)C1 + w(am -an)C2 ->0.
(f) tf {an} and {^nlare nuM sequences, so are {an + bn} and {an — bn}. If
further {an} is a null sequence and {bn} a bounded sequence, then
{anbn} is a null sequence.
Proof The first two assertions follow from
0 < w(an ± bn) < w(an) + w(fcnH0.
For the third assertion, w(bn) < C for all n, and hence, as n-^ oo,
0 < w(anbn) < wWwffcJ < w(an)C-^0.
3 The ring {K}w
Denote now by {K}w the set of all fundamental sequences
{an} = {al9a2,a39.. .}
where aneK.
Definition For any two sequences {an} and {bn} in {K}w, their sum
{an} + {bn}9 their difference {an} — {bn}9 and their product
{an} {bn} are defined by
K) + {bn} = K + bn}9 {an} - {bn} = {an - bn}9
and {an}{bn} = {anbn}.
By (e), these new sequences are again fundamental sequences and
hence lie in {K}w.
By this definition, {K}w becomes a commutative ring. Its identity
element under addition is the sequence
{0} = {0, 0, 0,...},
and its identity element under multiplication the sequence
{1} = {1,1,1,...}.
The ring {K}w is not a field because it contains divisors of zero, e.g.
{1,0,0,0,...}{0, 1,0,0,...} = {0}.
For every a in K the fundamental sequence
{a} = {a, a, a,...}

22 Pseudo-valuations and valuations
lies in {K}w. Hence {K}w contains a ring isomorphic to K as a subring.
Of particular importance is the set, P say, which consists of all null
sequences. By (b), such null sequences are also fundamental
sequences; hence P is a subset of {K}w.
In fact, P is an ideal in {K}w. For if {an} and {bn} lie in P, then, by (f),
so also {an} ± {bn}; and if {an} belongs to P, and {bn} is any element of
{K}w, then, by (a), {bn} is bounded, and therefore, by (f), {an} {bn} lies
in P. These two properties, however, define P as an ideal in {K}w.
All this is true when w is any pseudo-valuation of K. Let us now
impose on w the condition that it is a valuation of K, and consider any
two sequences {an} and {bn} in {K}w which are not null sequences,
hence do not belong to P.
There exist then, by (d), four positive numbers c, c', N9 and N' such
that
w(an)>c if n>N9 and w(bn)>c' if n>N'.
Hence, if N* is the larger one of N and N\ then
w(an)>c and w(bn)>c' if n>N*.
Since w is a valuation, this implies that also
w(anbn)>cc' if n>N*.
It follows that {an} {bn} = {anbn} is not a null sequence and so does not
lie in P. Therefore a product of two elements of {K}w can only then
belong to P if at least one factor lies in P. This proves the following
result.
(g) If w is a valuation of K, then P is a prime ideal
We stress that this property need not be true when w is only a pseudo-
valuation.
4 Residue classes (mod P), and the ring Kw
Assume again for the present that w is a pseudo-valuation of K.
Two sequences {an} and {a'n} in {K}w are said to be congruent
modulo P,
K) = {<} (mod p\ or simply, {an} = {<},
if the difference sequence {an} — {a'n} = {an — a'n} lies in P. Here the
sign = defines an equivalence relation since the following three proper-

Residue classes {mod P\ and the ring Kw 23
ties are easily proved,
(i) R} = R}-
(ii) If R} = R}> then also R) = R}-
(iii) If R} = R) and R) = {<}, ^ also R} = R'}.
As one usually does with equivalence relations, partition the ring
{K}w of all fundamental sequences into congruence classes (mod P)
by putting into the same class all those fundamental sequences that
differ only by null sequences. Let us call the set of all such residue
classes
Kw = {K}w/P-
Denote by A and B any two residue classes in Kw9 say with the
fundamental sequences {an} and {a'n} in A and {bn} and {b'n} in B. By
(f), the right-hand sides of the equations
R + K) - R + K) = R - <} + R - K)>
{an - bn} - R - b'n} = R - a'n) - {bn - Vn}9
RU - {<K} = R - a'n) {bn} + {<} {bn - b>n)
are null sequences since {an — a'n} and {bn — b'n} are null sequences.
These equations imply therefore the following three congruences,
{an + bn} = {a'n + K), {an - b„) = {a'n - b'n}, {anbn} = KKl
Hence we may define sum, difference, and product of residue classes
(mod P) as follows.
Definition If R} is any fundamental sequence in the class A and
{bn} any fundamental sequence in the class B, denote by A + B the
class containing {an} + R}, by A — B the class containing
R} — R}, and by AB the class containing {an} {bn}.
These new classes are well defined because, by what was just been
proved, A + B, A — B, and AB do not depend on the special
fundamental sequences used to fix the classes A and B.
With this definition, also Kw becomes a commutative ring.
Next let a and b be any two elements of K. The fundamental
sequences
{a} = {a, a, a,...} and {b} = {fe, fe, fe,.. .}

24 Pseudo-valuations and valuations
lie in {K} w. In the ring Kw we denote by (a) the residue class of {a}, and
by (ft) the residue class of {b}. It is then obvious, by the definition just
given, that
(a) + (b) = (a + ft), (a) - (b) = (a- b\ (a) (b) = (ab).
Hence Kw contains a subring which is isomorphic to K. For
convenience this subring is identified with K, and we do not
distinguish between the element aofK and the residue class (a) in Kw.
The identity element of Kw under addition is (0) which is the set P of
all null sequences; and the identity element under multiplication is (1).
5 When K is a field and w a valuation, then also Kw is a field
If there is no restriction on K and w, the residue class ring Kw may
contain divisors of 0 and so certainly not be a field. An example will be
formed by the #-adic ring studied in the next chapter.
There is, however, one important case when Kw becomes a field,
(h) When K is a field and w a valuation, then also Kw is a field.
Proof Let A be any residue class in Kw distinct from the zero class
(0) = P, and let further {an} be any fundamental sequence in A; hence
{an} is not a null sequence. Therefore, by (d), there exist two positive
numbers c and N such that
w(an) > c if n>N.
Define a new sequence {a*} by putting
* _J° if l<n<N-\,
a*~{l/a* if n>N.
This sequence is a fundamental sequence. For if both m>N and
n > AT, by the valuation property of w,
(1 1 \ w(aw — a„) ~
0<w(a*-a*) = w = ; w /;<c-2wam-an,
V*m aj w(am)w(an)
whence
lim w(a*-a*) = 0.
m -»oo
«-*■ oo
Denote then by ^4-1 the residue class (mod P) of {a*}. Then

The limit notation 25
AA~ x =(1) because
N — 1 zeros
K)K) = (0,...,0,1,1,1,...},
where the fundamental sequence on the right-hand side differs from
{1} by the null sequence
N-1 -ls
{-1,..., -1,0,0,0,...}.
Hence every element A ^ (0) of Kw has a reciprocal A~ *, thus proving
that K is a field.
This proof does not work when w is only a pseudo-valuation.
6 The limit notation
It is convenient to have a short notation for the residue class A
(mod P) which contains a given fundamental sequence {an}. We write
A = w-lim an.
n -*■ oo
Thus, by way of example, if {an} is a null sequence,
w-lim an = (0),
n-+ oo
and if a is any element of K, and
lim w(aB — a) = 0,
n-» oo
then evidently,
w~liman = (a).
n -*• oo
From the definition in § 4, and from (h), the following result holds.
(i) U
A = w-liman and B = w-lim bn,
n-+ oo n -*• oo
then also
A + B= w-lim (an + fen), A — B = w-lim (an — fen),
/r-* oo n -* oo
AB = w-lim anbn.
n -*■ oo
// further K is a field and w a valuation, and the fundamental

26 Pseudo-valuations and valuations
sequence {bn} is not a null sequence, then
A/B = w-lim(an/bn)
n-+ oo
where the finitely many suffixes n for which bn = 0 are omitted.
In the special case when K is the rational field Q, and w(a) = \a\ is
the ordinary absolute value, Kw becomes the field of real numbers,
and our limit notation becomes identical with that of real analysis.
Naturally, this was the reason for choosing this notation.
7 The extension of w to K
w
In general K is a proper subring of Kw, and therefore the pseudo-
valuation or valuation w is originally not defined on the whole of Kw.
We shall now give a meaning to w(A) when A is any residue class in
Let {an} be a fundamental sequence in K and
A = w-lima
n
n-+ oo
the corresponding residue class (mod P) in Kw. If A = (a) lies in X,
then {an — a} is a null sequence and therefore
lim w(an — a) = 0.
n-+ oo
This implies, firstly, that
w(an) = w(a + (an — a)) < w(a) + w(an — a) and therefore
lim w(an) < w(a\
n-+ oo
and secondly that
w(a) = w(an — (an —a))< w(an) + w(an — a) and therefore
w(a) < lim w(an).
n -*■ oo
Hence
w(A) = w(a) = lim w(an) if A = (a) lies in K.
n -*■ oo
Consider now the case when the class A is not of the form A = (a)
and thus does not belong already to K. We now define w(A) as
follows.

w(A) *s a pseudo-valuation or valuation 21
Definition If {an} is any fundamental sequence in A, put
w(A)= lim w(an).
«-*• 00
Two conditions have to be satisfied if this definition is to be valid: (i)
the limit w(A) must exist, and (ii) this limit may not depend on the
special fundamental sequence {an} used to fix the class A.
The first condition is satisfied because by formula (4) of §1
\w(am)-w(an)\<w(am-an%
which implies that the real sequence {w(an)} is a fundamental
sequence relative to the absolute value and so has a limit in the real
sense.
Next, if both fundamental sequences {an} and {a'n} lie in A, then
{an — a'n} is a null sequence and therefore, again by (4),
0 < lim |w(an) — w(a'n)\ < lim w(an — a'n) = 0,
n -*■ oo n -*■ oo
whence
lim w(an) = lim w(a$.
n -*■ oo n -*■ oo
8 w(A) is a pseudo-valuation or valuation on Kw
It will next be proved that w(A) is a pseudo-valuation on Kw9 thus has
the following three properties.
(1) w(0) = 0, w(A) > 0 if AeKw is distinct from (0) = 0.
(2) w(A ±B)< w(A) + w(B) if A and B lie in Kw.
(3) w(AB) < w(A)w(B) if A and 5 lie in Kw.
Let the residue classes A and .B be defined by the fundamental
sequences {an} and {fen}, respectively.
Firstly, if A = 0, then {an} is a null sequence and therefore w(A) = 0;
if, however, ^4/0, then {an} is not a null sequence, and so, by property
(d), in § 3, there exist two positive numbers c and N such that w(an) > c
for n>N, and hence also w(A) > c> 0.
Next, by the properties of real limits,
w(A ±B)= lim w(an ± bn) < lim (w(an) + w(fen))
n -*■ oo n -*■ oo
= lim w (an) + lim w(fen) = w(A) + w(B),

28 Pseudo-valuations and valuations
and similarly,
w(AB)= lim w(anbn)< lim w(an)w(bn)
n -*■ oo n -*■ oo
= lim w(an) lim w(bn) = w(A)w(B\
n -*■ oo n -*• oo
which proves also the properties (2) and (3).
In the special case when w is a valuation of K, in the last formula
w(anbn) = w(an)w(bn% and the same proof shows that
w(AB) = w(A)w(B),
so that w(A) is also a valuation on Xw.
Similarly, if w(a) is non-archimedean on K, then
lim w(an + bn) < lim max (w(an\ w(bn))
n -*■ oo n -*■ oo
< max ( lim w(an), lim w(ftn) I = max (w(A% w(B)).
\n -*■ oo n -*• oo J
whence
w(^ + B) < max (w(^), w(5)),
so that w(A) is also non-archimedean on Kw.
We call the pseudo-valuation or valuation w(A) the continuation of
w(a) to Kw.
9 The approximation of elements of Kw by elements of K
When K = Q is the rational field and w(a) = |a| is the ordinary
absolute value, which is a valuation on K, then Kw becomes the field
of real numbers R, with w(A) = \A\ the absolute value on R. Every
element A of KW = R can be arbitrarily well approximated by
elements of K = Q. For this purpose write simply A as an infinite
decimal fraction and break off this fraction after a sufficiently large
number of digits.
We prove now that also in the general case the elements of K lie
dense in Kw. For this purpose let
A = w-liman
«-*• 00
be an arbitrary element of Kw defined by means of any fundamental
sequence {an} in the residue class A. As m runs over the positive

X as the completion of K 29
integers, (am) is by definition the residue class of the special
fundamental sequence
and therefore A — (aJ = A-am can be defined as the w-limit
A - am = w-lim (an -am) (m = 1, 2, 3,.. .).
n -»■ oo
The definition of w(A) in §7 implies therefore that
w(A - am) = lim w(an - am) (m = 1, 2, 3,.. .).
«-*• 00
Here {an} is a fundamental sequence, whence
lim w (A- am) = lim w(an - am) = 0,
m -*■ oo m -*■ oo
«-*• oo
which proves that for sufficiently large m the number w(A —am) is
arbitrarily small. Thus, relative to w, the elements of K are everywhere
dense in Kw9 just as the elements of Q were in i^ relative to the absolute
value \a\.
10 Kw as the completion of iiT relative to w
As constructed, the ring Kw has the pseudo-valuation (or possibly the
valuation) w(A) which continues w(a) from the subring K to Kw. We
can therefore now define bounded sequences, fundamental sequences,
and null sequences in Kw9 in analogy to the definitions in §3, so
perhaps coming to an extension ring of Kw which we might denote by
(Kw)w. As will now be proved, such a repetition of the construction is
unnecessary because it leads to nothing new.
For let
\An} = \Al9 A2-> A3,.. .}
be any fundamental sequence relative to w(A) formed by elements of
Kw. We shall prove that there exists an element A of Kw such that
lim w(An — A) = 0.
n -*• oo
To begin with, the fundamental sequence {An} has the property
lim w(Am-An) = 0.
m -*■ oo
n -*• oo
Next, by the last section, to every element An of {An} there exists an

30 Pseudo-valuations and valuations
element an of K such that
*>(An-an)<\/n.
Hence {An — an) is a null sequence and so, by property (b) in § 3, also a
fundamental sequence. The sequence
K) = (4.} ~ (4, ~ an)
is therefore likewise a fundamental sequence in Kw9 and it is in fact a
fundamental sequence in K because its elements an lie in K. It has then
a limit
A = w-lim an
«-*• 00
in Xw. Moreover, both
{^-aj and {^n-an}
are null sequences in Kw9 and hence their difference
{A - An} = {A- an) - {An - an}
also is a null sequence in Kw. This implies that
lim w(A — An) = 0.
n -*• oo
It follows that the fundamental sequence {,4} = {A, A, A, .. .}
in Kw has the same w-limit in (Kw)w as the sequence {An} =
\Al9 A29 A$9 . . .},
A = w-lim An in (KJW.
«-*• 00
But ^4 lies already in Xw, and hence (Xw)w is identical with Kw.
On account of this important property Kw is called the completion
of K relative to w, or the w-completion ofK.
Thus, for example, R is the completion of Q relative to the absolute
value. In the next chapter, we shall similarly consider the completions
of Q relative to the different g-adic values, in particular the p-adic
values.
11 Problems
1 Let {an} = {au a2, a3,...} be a fundamental sequence with respect
to the pseudo-valuation w of K. Let further mu m2, m3,... be a
strictly increasing sequence of positive integers. Prove that the

Problems 31
sequence {amn} = {ami, am2, am3,.. .} is a fundamental sequence
with the same w-limit as {an}. (Suggestion: show that the difference
of the two sequences is a null sequence.)
2 If w is non-archimedean, prove that {an} is a fundamental
sequence if and only if the sequence {ax —a2,a2 —a3,a3 — a4,...}
is a null sequence.
3 K is a ring, and w0 is defined by
w0(0) = 0, and w0(a) = 1 if aeK and a =£ 0.
Show that w0 is a pseudo-valuation, and determine all the
corresponding bounded sequences, fundamental sequences, and
null sequences, and hence find KWQ. Under which additional
conditions is w0 a valuation?
4 If wx (a) and w2(a) are two pseudo-valuations of K, prove that also
W(a) = Wl(a) + w2(a) and W*(a) = max(w1(a), w2(a))
are pseudo-valuations of K. Show that W and W* have the same
bounded sequences, the same fundamental sequences, and the
same null sequences.
5 Let Z be the ring of rational integers. Put v(0) = 0. If further a e Z
does not vanish, and n is the largest positive integer such that n! \a,
put v(a) = \/n. Show that v(a) is not a pseudo-valuation of Z,
determine which properties of a pseudo-valuation are satisfied by
v9 and prove that v(ab) <max(y(a), v(b)).
6 Let Kbea ring, w a pseudo-valuation of K, x an indeterminate,
and K[x] the ring of all polynomials
A =a0 +alx + ... +arxr
in x with coefficients in K. Put
W(A) = w(a0) + w(ax) + ... + w(ar)
and
P^*(^4) = max(w(a0), w(ax),..., w (a,)).
Show that both W and W* are pseudo-valuations of K[x], and
that W*(A) is non-archimedean if w(a) is non-archimedean.
7 Let w be a pseudo-valuation of K, and c> 1 and C > 0 be real
constants. Prove that also cw is a pseudo-valuation, and that if w is
non-archimedean, then wc is a pseudo-valuation.
8 Let C be the complex number field, x an indeterminate, and C(x)

32 Pseudo-valuations and valuations
the field of rational functions in x with coefficients in C. Further let
c be any complex number. To every rational function r =f= 0 in C(x)
there exists a unique integer f such that (x — c)fr(x) is finite and
not 0 at x = c. Put
wc(0) = 0 and wc(r)=ef if r =/=0.
Prove that wc is a non-archimedean valuation of C(x).
9 Let the notation be as in problem 8. Every r =j= 0 in C(x) can be
written in a unique way as the quotient r = p/q of two polynomials
without common zero where q is monic. Put
w*(0) = 0 and w*(r) = edegreeof^-degreeof<? if r=j=0.
Show that w* is a non-archimedean valuation of C(x).
10 Let the notation be as in problems 8 and 9. Construct a sequence
of elements of C(x) which is a fundamental sequence with respect
to wc, but not to w*, and also a sequence which is fundamental
with respect to both wc and w*.

3
g-adic and p-adic numbers
1 The g-adic ring Qg and the />-adic field Q
Denote by g > 2 any integer, and by p any prime. The rational field Q
has the absolute value \a\ which is a valuation of Q, and the
completion of Q with respect to \a\ is the real number field R. But, in
addition, Q has also all the g-adic pseudo-valuations \a\g and all the
p-adic valuations \a\p as defined in Chapter 1.
Denote by Qg and Qp the completions of Q relative to \ag\ and to \a\p9
respectively. Then Qg is a ring and Qp a field. We shall soon prove that
Qg is not also a field if g has at least two distinct prime factors; for then
we shall construct divisors of zero.
Qg is called the g-adic ring, and its elements are called g-adic
numbers; similarly Qp is called the p-adic field, and its elements are
called p-adic numbers. We shall obtain a better understanding of these
numbers by expanding them into a simple kind of series, of the type
already considered for rational numbers in Chapter 1.
2 Convergent series in K
Let K be again any ring with a pseudo-valuation w(a) and the
corresponding completion Kw. In analogy to what one does in real
analysis, it is often convenient to define elements of Kw not as the
limits of fundamental sequences, but as the sums of infinite series
Ai+A2+A3 + ...= £ An
n> 1
where the terms An lie in K or more generally in Kw.
Such a series is called w-convergent, with the w.-sum S, if the
sequence
n
{SJ = {Si, S2, S3,...}, where Sn= £ Ak
*= i
(?i = 1, 2, 3,...)

34 g-adic and p-adic numbers
is a fundamental sequence relative to w, with the w-limit
S = w-lim S.
We then write
«-*• oo
00
S = w-£ A-
n= 1
The necessary and sufficient condition for this convergence is that
lim w(Sm - Sn) = 0.
m-+ oo
n -*• oo
Since w(Sm — £„) = w(Sn — iSm), we may without loss of generality
assume that m> n, hence that
m
k = n+ 1
This leads immediately to the following convergence test.
The series £ An is w-convergent if and only if
n> 1
lim w(An+1+An + 2+... + AJ = 0. (1)
m -*• oo
«-*• oo
In the real field, this becomes the usual Cauchy condition for
convergence.
In particular, on taking m = n + 1, it follows that
lim w(^m) = 0 (2)
m -*• oo /
is a necessary condition for convergence. However, it is wot in general a
sufficient condition, as we see when K = Q and w(a) = |a|, for then the
harmonic series
n> 1 n
satisfies (2), but is known to be divergent.
There is one important case in which the condition (2) can be shown
to be also sufficient.
Let wbea non-archimedean pseudo-valuation ofK. Then £ An is
n> 1
w-convergent if and only if (2) is satisfied. (3)
Proof. It only remains to be shown that this condition is sufficient, and

Reordering of the terms of a series 35
this follows from
w(An + ! + An + 2 +... + Am) < max (w(An + ±)9 w(An + 2),.. ., w(^J).
Thus, in particular, since \a\g and \a\p are non-archimedean, it
follows that
every infinite series in Qg or Qp converges if and only if its terms
form a null sequence. (4)
3 Reordering of the terms of a ^-convergent series
Consider first again the case when the ring K has a non-archimedean
pseudo-valuation w. We shall often have opportunity to apply the
following result.
00
tfw~ X An exists, and if £ A'n is a series with the same terms,
n = 1 n > 1
oo
but in a different order, then also w-£ A'n exists and is equal to
n= 1
oo
w-IX. (5)
n= 1
Proo/ Denote by e an arbitrarily small positive constant and by N so
large a positive integer that
w(An) < e and w(A'n) < e for n > N
and that further
/ oo N \
w(w- X 4,- Z 4,)<e. (6)
\n=l n=l /
Put
5= J An and 5'= £ ^,
n= 1 n= 1
and further denote by Sx and Si the sums of all the terms of S for
which
w{An) > e,
and of all the terms of S' for which
wK)>e,
respectively. It is clear that S1 and Si have the same terms, hence that

36 g-adic and p-adic numbers
Next, S differs from Sl only by terms satisfying
w(An) < e,
and 5" from Si only by terms such that
w(A'n) < e,
It follows therefore that
w(5-51)<e and w(S'-Si)<e,
hence also
w(S - S') < e.
On combining this with (6) and the definition of 5", we see that
/ oo N \
\n=l n= 1 /
and as e tends to 0 and N to oo, we obtain the assertion. In particular,
The terms of every convergent series in Qg or Qp may be reordered in
any way without changing its convergence or its sum. (7)
The results (5) and (7) are quite different from those in real analysis.
There the terms of a series may only then be reordered without
changing its convergence or its sum if the sum of the absolute values of
the terms is itself convergent, a theorem due to Dirichlet.
The property (7) is the more surprising because, similarly as in the
real case, the following result holds.
There exists a convergent series £ An in Qg such that the real
n> 1
00
series £ \An\g diverges. (8)
n= 1
Proof. Choose as the consecutive terms of the series
1; #, g times repeated; g2, g2 times repeated; #3, g3 times repeated; etc.
These terms tend to 0, and so the series converges. On the other hand,
00
Z \An\g = i +q-q~1 +g2-g~2 + g3-g~3 + ... = 00.
n= 1
4 The canonic expansion of a g-adic number
According to the notation of Chapter 2 the #-adic and p-adic limits
should be denoted by |a|g-lim and |a|p-lim, respectively. This notation

The canonic expansion 37
is rather clumsy, and we therefore use from now on the simpler
9 P
lim and lim.
We shall further denote the sum in Qg or Qp of £ An by
n > 1
00 00
X An (gf) and £ A« 0>X
n = 1 n = 1
respectively.
In this notation, every element A of Qg may be defined as the limit
9
A = lim an
n -*■ oo
of a suitable fundamental sequence {an} in Q relative to \a\g. Here, by
Chapter 2, § 7,
|4|,= lim |ajg.
n -*■ oo
We can have ^ = 0 only if {a„} is a null sequence. Let this case be
excluded; then \A\g ^ 0 and therefore
\A\g = gf,
where /is a certain integer (> 0, < 0, or = 0), As is set as problem 1,
Chapter 2 § 11, the fundamental sequence {an} may be replaced by
any infinite subsequence without changing its limit, Hence we are
allowed to assume that this sequence has the following two
properties,
— nf
an\a = 9J (n=l, 2, 3,...); (9)
\am-an\g<g-N if nhn>N (N = 1, 2, 3,...). (10)
Each an is a rational number. By Chapter 1, § 5, it can be written in
the form
<*n=<*n,-f9~f+<*n,-f+l9~f+1+'..+<*n,N-l9N~1
+ 9 (Rn, N/Sn, Nh
where the coefficients
an, - /' an, - f + 15 * • * •> an, N - 1
are digits 0,1,..., g — 1, with in particular

38 g-adic and p-adic numbers
and where further Rn N and Sn N are integers satisfying
(Rn N, Sn N) = (g, Sn N) = 1.
There is a similar representation
am=am, - fG +am,-f+lQ + • • • + am, N - 1 G
+ GN(Rm,N/Sm,N)
for am.
It follows that
R
k=-f \^m,N
um
\, N
Here
\G> ^m,N^n,N)= ^
hence for ra, n>N, by (10),
N - 1
l^m ^nlgf
I (am,k-an,k)Gk
k= -f
<G~". (11)
- N
9
In the sum, all the factors
am, k an, k
are differences of digits 0, 1,..., g — 1 and hence satisfy
-{g-V)<amk-ank<g-\.
They can thus only then be divisible by g if
am, k = an, k'
On considering successively in the sum in (11) the terms divisible
exactly by g~ f, g~ f + \..., gN ~ \ respectively, they all are
necessarily equal to 0, and hence it follows that
where for m,n>N the new digits ak no longer depend on the suffixes m
or n.
This result is valid for all N, and it follows in the limit as N -► oo that
the #-adic number A can be written as the convergent series
A =a_fg-f + a_f+lg-f + 1+a_f + 2g-f + 2 + ... fer), (12)
where the coefficients ak are digits. If A ^0 and \A\g = gf, then
a_fi=0;
if, however, A = 0, then all the digits are equal to 0.

g-adic and p-adic integers 39
This series for A is called the canonic series of A.
The canonic series of A is unique. (13)
It suffices to prove this when A =£ 0. A second canonic series would
begin with the same power g~f of g and have the form
A=a'_fg-f+a'_f + 1g-f+1+a'_f + 2g-f + 2 + ...(#), (14)
where also
a,- f, a~/ + 1? a '_ f + 2,. •. are digits and a'_ f^=0. We assert that
ak=a'k (k= -f -/+1, -/ + 2,...),
so that the two developments for A are identical. If this were false,
there would be a smallest suffix r such that
ar ^=a'r.
Then on subtracting the two series (12) and (14), it would follow that
0 = (ar — a'r)gr + terms in higher powers of g,
which is evidently impossible.
The g-adic expansion of rational numbers studied in Chapter 1 is a
special case of the expansion (12) of any element A oiQg. As we found,
a canonic series represents a rational number if and only if it is periodic.
Also in the case of the general canonic series in Qg we shall follow
Hensel's notation and write (12) in the abbreviated form
<**- -— a_ ^a — r _\_ ^a_ r _\_ 2• • *^o> ^ia^a^• • • \gu
where the comma stands again between a0 and ax.
What has been proved for general g carries over without change to
the case when g = p is a prime. We then obtain the canonic series of a
p-adic number.
5 g-adic and /i-adic integers
A g-adic number A is called a g-adic integer if
\A\g <1,
and similarly for p-adic integers. The two sets of g-adic integers, Ig say,
and p-adic integers, Ip say, both form rings. For if, for example,
\A\g <1 and \B\g < 1,
then by the properties of a non-archimedean pseudo-valuations also
\A+B\g<l, \A-B\g<l, and \AB\g<l.

40 g-adic and p-adic numbers
The canonic series of a g-adic integer has the special form
A=a0+a1g + a2g2 + .. ■ =a0,a1a2a3... (g).
As a special case, such an expansion holds for p-adic integers.
All rational integers are also g-adic and p-adic integers; thus Z is a
subset of both Ig and Ip.
If the g-adic number A is not a g-adic integer, thus if
\A\g = gf where f> 1,
then, on putting
R = gfA9 S = gf,
A becomes the quotient
A = R/S
of two g-adic integers. In the language of algebra this means that Qg is
the quotient ring ofIg. On the other hand, Qp is the quotient field ofIp
because it is a field.
A g-adic unit is any g-adic number A satisfying the two relations
\A\g=l and 1^-^ = 1.
As will soon be proved, this means that the canonic series of A has the
form
A=a0,a1a2a3.. .(g) where (g, a0) = l.
When g is a prime p, the condition for a0 takes the simpler form
a0 =^ 0.
6 Problems
1 Show that
9
lim an
n -» oo
exists if and only if
9
lim (an +1 - an) = 0.
n -» oo
2 Let {an} be a sequence of p-adic numbers (p prime) such thatan ^ 0
for all n, and let bn = a1a2. ■ .an.

Problems 41
(i) Prove that
00 p
[] an = lim bn
n = 1 n-> oo
exists and is distinct from 0 if and only if
p
lim a„ = 1.
n -*■ oo
(ii) Show that the analogous #-adic result does not hold if g is
divisible by at least two distinct primes,
3 Show that every a ^= 0 in Qp (p a. prime) can be written as a
convergent infinite product
00
a = pfa0 x 11 (l+anpn) (/?),
n= 1
where f is a rational integer, a0, al9 a29... are digits 0, 1,...,
p — 1, and #q ^ 0.
4 Decide whether the following limits exist, and determine them if
they do.
9 0 gn +a
lim n\: lim where b is not a zero divisor in Qn, i.e. be ^0
_* ^ n^ -\- h 9
n -* oo w -► oo »* • i t/
6
for all c ^ 0 in Q^; lim 43n.
« -> 00
5 Which of the following p-adic series are convergent (p prime) ?
v I v pn V fnl\ V ^
«>1 «>ln! n>l\n/ n>ln~T~l
6 Let 0 = 99, and let S = £^ x w! (99). Show that |S|99 = 1/99.
7 Find the sums of the following #-adic series.
00 00 00 00
E «xn!(g); X g"; £ n9"l I(-l)n«V-
«=1 «=1 n= 1 «=1
8 Determine the canonic series of ^, y, ^, and |, for p = 2, and hence
that of^ + ^ + ^ + i
9 Determine the six first digits of the 3-adic number Yj7= i n ••
10 Show that the #-adic infinite product Y\„= i (1 + 02 ) (o) converges
and that it is a rational number.

4
The arithmetic of Q and Qp
Arithmetic operations (addition, subtraction, multiplication, and
division) in the real field R are usually carried out using decimal
fractions. We shall for the same purpose in Qg and Qp apply the
canonic series and Hensel's abbreviated form of the same. As was
shown, every #-adic number A ^= 0, say with \A\g = gf, can be written
as
A=a_fg~f+a_f + 1g~f+1 + ...+a_1g~1
+ a0 +axg +a2g2 + ...
where the coefficients an are digits 0,1,..., g — 1, with a_f^0. In
particular, if A is a #-adic integer, the terms in negative powers of g are
missing, and
A=a0+a1g +a2g2 + ... =a0,a1a2. •. (g).
Here, for \A\ < 1, one or more of the first digits a0, al9a2,.. - may
vanish.
If (j) is any negative or positive integer, the abbreviated form of the
canonic series of g* A is obtained by moving the comma |</>| steps to
the right if <f> < 0, and <f> steps to the left if </> > 0. It suffices therefore
often to consider only #-adic integers.
The arithmetic of p-adic numbers is simply the special case of #-adic
arithmetic when g = p is a prime.
1 Reduction of a general series to canonic form
Before studying the arithmetic in Qg, it is convenient first to prove a
simple reduction theorem for a class of series more general than the
canonic one.

Reduction to canonic form 43
Let
00
n= ~f
where the coefficients un are rational integers which need not be digits.
Since _n
|wn#n|,<0 n,
this series converges. The following construction allows us to express
it as a canonic series.
This construction is based on the identity
ungn = (un-gvn)gn + vngn+1.
Here vn can be chosen as a rational integer in exactly one way such
that
0<un-gvn<g-l,
thus that un — gvn becomes a digit 0, 1,..., g — 1.
The reduction of the series for A now runs as follows.
First choose the integer v_f such that
u_f-gv_f,=a_f say,
is a digit and apply the equation
u_fg~f + u_f + 1g~f+1 =a_fg-f + (w_/ + 1 + v_f)g~f+1.
Secondly choose the integer v_f + 1 such that
(u_f+1+v_f)-gv_f + 1,=a_f + 1 say,
is a digit and apply the equation
(u_f+1+v_f)g-f + 1+u_f + 2g-f + 1
= a_f+1g~f + 1 +(u_f + 2 + v_f+1)g-f + 2.
Thirdly choose the integer v_f + 2 such that
(u_f + 2 + v_f+1):-gv_f + 2,=a_f + 2 say,
is a digit and apply the equation
(u_f + 2 + v_f+1)g-f + 2 + u_f + 3g-f + 3
= a_f + 2g-f + 2 + (u_f + 3 + v_f + 2)g-f + 3.
Continuing in this way, determine the consecutive digits
it is then clear that A has the canonic series
00

44 The arithmetic qfQ and Q
9
The same kind of reduction can even be applied when the
coefficients un are not rational integers, but are either rational
numbers satisfying
K\g<l
or more generally are #-adic integers in Qg. In the first case one applies
Lemma 1 of Chapter 1 § 3, with G = g, and in the second case one
replaces un by the first digit of its canonic series.
2 Addition and subtraction
Suppose now that A and B are two #-adic numbers satisfying, say,
\A\g>\B\g and \A\g = g'.
Let their canonic series be
00 00
A= t anGn to) and B= £ bngn {g\
n=-f n=-f
where thean and bn are digits. Here a _ f^0, but it is possible that one
or more of the first digits b_ f, b_ f + l9 b_ f + 2,... are equal to 0.
For A ± B we obtain immediately the convergent series
00 \
a±b= ^ K±K)Gn (g),
n= ~ f
which, however, in general will not be canonic. It can immediately be
changed into a canonic series by means of the reduction process just
discussed.
By way of example, let
00
A = l, B= ^ (g-l)gn (g).
n = 0
On carrying out the reduction, we find then successively that
00
A + B = g+ ^ (0-1)0"
n = 1
oo
= 0+0-01 + Z (0-1)0"
n = 2
00
= o + oV+0-02+ Z (0-1)0",
n = 3

Addition and subtraction
45
etc., and on repeating this indefinitely,
00
A + B= ^ 0-0- = 0(0).
n = 0
Hence
00
B= £ (g- 1)3» = - 1 (g),
n = 0
an equation obtained in a different manner in Chapter 1 § 7.
In the same way we find that
00
0-<T0+ Z (g-l)gn = 0(g).
n= -<f>+ 1
Hence if the B above satisfies \B\g = g^, so that b_(j) is its first digit
distinct from 0, then
00
-B = (g-b_<p)g-*+ £ (g-bn- \)g"(g)
evidently is the canonic series for — B. Instead of subtracting B from
A, we may add — B to A using this series.
Naturally also more than two #-adic numbers may be added, and
the same reduction process allows us to find their sum. As an example,
let us find the first eleven digits in the 2-adic series for
o _ 2 , 4 ■ 8 ■ 16 ■ 32 , 64 , 128 , 256 , 512 , 1024
^ — 1+2 + 3+4+5 + 6-1- ? -f- g -f- 9 -f- 10.
An easy calculation gives the abbreviated forms of their canonic series
for the successive terms as
0,1000000000
+ 0,1000000000
+ 0,0011010101
+ 0,0100000000
+ 0,0000101100
+ 0,0000110101
+ 0,0000001110
+ 0,0000100000
+ 0,0000000010
+ 0,0000000010
2)
2)
2)
2)
2)
2)
2)
2)
2)
2)

46 The arithmetic ofQg and Q
so that, on simply adding digits in the same column, the sum becomes
0,21113 22432(2).
On reducing this, it follows finally that, apart from an error not
greater than 2~ 10,
5 = 0,0000000001(2).
In fact,
1024 x 73
S =
32 x 5 x 7
3 Multiplication
Multiplication can be carried out in almost the same way. Let
OO 00
A= £ a„gn(g) and B= £ bjfig)
n= - f n= - <f>
be two g-adic numbers in canonic form where
\A\g = gf and \B\g = g*9
so that the first digits a _ f and b_(f) are distinct from 0. On multiplying
the series term by term and rearranging the terms, we find that
00
AB= ^ un9n(G\
n= ~ f~4>
where
u- f-<t> + l = a- f + r°-4> +a-/k-</, + i>
etc. This series is in general not canonic, but again the method of §1
allows us to reduce it to canonic form.
As an example, let us find two 6-adic numbers not 0 the product of
which vanishes; thus the two 6-adic numbers are divisors of 0. We
may assume that both numbers are 6-adic integers, thus of the form
OO 00
A= Z a„6" (6) and B= £ b„6" (6).
n=0 n=0
Here the coefficients are digits 0, 1, 2, 3, 4, or 5. Since
\AB-a0b0\6<%,

The reciprocal of a g-adic number 47
the hypothesis that AB = 0 implies that a0b0 must be divisible by 6.
This condition is satisfied if
a0 = 2 and b0 = 3.
Then
00
where
AB= ^ "„6"(6)
n = 0
w0 = 6,
ul = 2bx + 3al9
u2 = 2b2 + ax b1 + 3a2,
w3 = 2ft3 + axfe2 + a2^i + 3a3,
w4 = 2ft4 + a1b3 + a2ft2 + a3^i + 3a4,
w5 = 2b5 + axft4 + a2ft3 + a3b2 + a4&i + 3a5, etc.
It is necessary that all the digits in the canonic series for AB are equal
to 0. We must therefore successively satisfy the following conditions
in digits, 6\2b1 + 3a1 + 1, solution al=b1 = l; 6\2b2 + 1 + 3a2 + 1,
solution a2=0, b2 = 2; 6\2b3 + 2 + 0 + 3a3 +1, solution a3
= 1, fe3 = 0; 6|2b4 + 0 + 0 + l + 3a4 + l, solution a4 = 0, fe4 = 2;
6|2b5+ 2 + 0 + 2 + 0 +3a5 + 1, solution a5 = 1, fe5 = 2, etc.
Therefore
,4 = 2,10101... (6) and B = 3,12022... (6).
On multiplying these two numbers we obtain
,4£ = 6,5555(5+ 6). ..(6),
which in reduced form is
AB = 0,00000... (6),
as required. There is no difficulty in evaluating more places of A and
B. The theory of zero divisors in Qg will be studied in the next chapter.
4 The reciprocal of a g-adic number
Let
A = a0,a1a2a3 .. .(g)
be a g-adic number satisfying

48 The arithmetic ofQg and Q
hence a0 ^ 0. The problem arises whether there is a reciprocal A ~ 1 of
A such that
AA~1 = 1.
If (a0, g) > 1, this may or may not be so. Thus 2 = 2,000 ... (6) has
the reciprocal \ = 30,000 ... (6). On the other hand, if A = 2,10101
... (6) and B = 3,12022 ... (6) are the two zero divisors given in
§ 3, then AB = 0, and so the equation AA~ 1 = 1 would imply that
ABA~ 1 =0.^4" * = B = 0 which is certainly false. Thus A has no
reciprocal.
For this reason assume that
(a0, 0)=1.
It is then possible to construct a reciprocal A'1 = c0, c1c2c3... of A.
This requires that
AA'1 = (a0c0), (a^Q + a0c^) (a2c0 +alcl+ a0c2)
(a3c0 + a2c1 + a1c2 + a0c3) (g)
is equal to 1. Digits c0,cu c2,... satisfying this condition can be found
as follows.
Since (a0, g) = 1, there is a digit c0 such that
a0c0 = l+gdl9
where d1 is a certain non-negative integer. There is secondly a digit cx
such that
a^Q + a^ + dx =-0 + gd2,
where d2 is a non-negative integer. Thirdly, there is a digit c2 such that
a2c0+a1c1+a0c2 + d2 = 0 +gd3,
where d3 is a non-negative integer. This construction can be
continued indefinitely and leads to the canonic series of A~ 1.
In the special case when g = p is a prime, it follows from a0 ^= 0 that
(a0, /?) = 1, hence that ^4 ^ 0 has a reciprocal. This is of course to be
expected since Qp is a field.
5 Division
Let again
A = a0, ala2a3... (g) and B = b0,b1b2b3.. .(g)

Division 49
be two g-adic integers written in the canonic form. The question arises
whether the quotient A/B exists, and if so, to determine its canonic
series.
As we know already from the last section, the existence of this
quotient is doubtful if (fe0, g) > 1; a full answer will not be given until
the next chapter. Let therefore
Then a5_1 exists such that BB~ 1 = 1, and the quotient is given by
A/B = AB~K
There is, however, a more direct method for finding A/B which
corresponds to the division of decimal fractions in the real field. This
method begins by constructing a table of the multiples
dB, where d = 0, 1,..., g — 1,
each multiple being written in the abbreviated canonic form.
First, since (fe0, g) = 1, there is a digit d0 such that
d0B = a0,... (g)
and therefore
A — d0B = 0, a[a2a'3... (g),
where al9 a2, a3,... are certain digits which can be calculated.
Secondly, there is a digit dl such that
dj$g = 0, a\.. . (g)
and therefore
A - (d0 + dyg)B = 0,0 a'^ ., . (g).
Thirdly, we can find a digit d2 such that
d2Bg2 = 0,0 a'2\ .. (#)
and hence
A-{d0 + dxg + d2#2)£ = 0,00 a(3).., (#)
In this way we can continue, making more and more digits of the
difference equal to 0; the result is that
A/B = d0,d1d2d3 ... (g).
By way of example, let g = 10 and
A = 1 - 1,000 ... (10), B = 1 = 7,000 ... (10).

50
The arithmetic qfQg and Q
For the multiples of B we obtain the table
0x7 = 0,0 1
1x7 = 7,0 1
2x7 = 4,1 1
3x7=1,2 I
4x7 = 8,2 I
5x7 = 5,3 (
6 x 7 = 2,4 1
7 x 7 = 9,4 il
8x7 = 6,5 1
9 x 7 = 3,6 J
:io),
(10),
;io),
;io),
;io),
:io),
(10),
(10),
(10),
(10).
Only the first digits after the comma have been given because all the
further ones are 0. Compare with the multiplication table in the real
case!
Since (10),999999. .. = 0 (10), we can write 1 = 1,(10)999999... (10).
The division runs now as follows.
7,000.. .|1,(10)9 99999... = 3,417582(10)
1 2
8999999...
82
799999...
70
99999...
94
5 999...
53
699...
65
49...
41...
o. . .

A solution of x2 + 1=0 in Q5 51
The canonic series for j is of course periodic, and the bar over the six
digits 417582 denotes that these form the period.
To verify this result,
7 x 3,417582... = (21),(28)(7)(49)(35)(56)(14)... = 1,000000... (10),
as follows by the reduction method of § 1.
6 A solution of x2 + 1 = 0 in Q5
As a further example in g-adic and p-adic arithmetic, let us find a
square root of — 1 in Q5, say
/i. —— @r\) U1U2W3. . . \3)'
On squaring, we find that
A2 + 1 = (#o + 1)> (2a0a1)(2a0a2 + al)(2a(/i3 + 2axa2)
(2a0a4 + 2axa3 + a\){2a0a5 -\-2axa4 + 2a2a3)... (5)
which must be made equal to 0.
Firstly, a\ + 1 is to be divisible by 5, which is true for both a0 = 2
anda0 = 3; let us choosea0 = 2. From here on all the further digits are
determined uniquely, and we find successively
22+ 1 = 5 = 0+1x5,
hence
ax = 1, so that 2^^ + 1 = 5 = 0+1x5,
a2 = 2, so that 2a0a2 + a\ + 1 = 10 = 0 + 2x5,
a3 = 1, so that 2a0a3 + 2axa2 + 2= 10 = 0 + 2x5,
a4 = 3, so that 2a0a4 + 2a1a3 +a\ +2 = 20 = 0 + 4x5,
a5 = 4, so that 2a0a5 + 2axa4 + 2a2a3 +4 = 30 = 0 + 6x5,
etc.
This gives
A = 2,12134... (5)
or if the process is continued a little further,
,4 = 2,1213423032...(5).
Naturally also
-,4 = 3,3231021412...(5)

52 The arithmetic ofQg and Q
is a solution. But since Q5 is a field, there are no further solutions. We
can check the calculations by forming
A2 + 1 = 5,4(9)(8)(18)(26)(29)(38)(31)(52)(48)... = 0,0000000000 (5),
as follows by the usual reduction.
7 Problems
1 Let A = 2,10101... (6) and B = 3,12022... (6) be the zero divisors
obtained in § 3. Show that in neither number are the digits
periodic.
2 Find the first five digits of the two 3-adic solutions of A2 — 7.
3 Show that there is no 7-adic number A — 2,a1a2a3.,. (7) such that
A3 + A2 - 2A - 1 = 0.
4 Is there a 5-adic number A — 3, aya2a3 ... (5) such that A3
+ 3,4 + 1=0?
4 42 43
5 Determine the first ten digits of 1 + — + — + — +... (2) and of
3 32 33
1+l!+2! + 3!+---(3)-
6 It has been found by A. C. M. van Rooij & W. H. Schikhof (1971)
that
{1,00110000101111...(2),
0,01210110202120...(3),
3,22240032010313...(5).
Verify these results!
7 Carry out the following calculations (Hensel 1913).
2,3102114 + 3,141202132 (5), 1,314 x 0,2103 (5),
3,12053 1 x 4,435024(6), 3,12 :4,21 (5).
8 Write in canonic form the 5-adic numbers
95 952 953 954
evaluating the first eight digits and showing that these converge.
9 Divide 1 by 2,12134 (5), and 2,12134 (5) by 3,3231 (5),
10 Find a pair of 10-adic numbers A = 29a1a2a3... (10) and B =
5, b1b2b3... (10) such that AB = 0.

5
The decomposition ofQg into p-adic
fields
In Chapter 4 § 3 an example of a pair of zero divisors in Q6 was
constructed, showing that Q6 certainly is not a field. In this chapter,
we shall study for general g > 2 the structure of Qg and establish a
result by Hensel (1913) that this ring is the direct sum of p-adic fields.
1 The special case when g is a power of a prime
As an example, let g = pr be the rth power of a prime p where r > 2. It is
easily shown that for all a in Q,
\a\p< \a\pr<pr~1\a\p. (1)
Hence a sequence {an} of elements of Q is bounded, is a fundamental
sequence, or is a null sequence, relative to \a\pr if and only if it is the
same relative to \a\ . The construction of the completion in Chapter 2
shows then the complete identity of Qpr with Qp.
It is in fact not difficult to express every pr-adic number A as a
p-adic number, and vice versa. For let
00
a= I An(Py
n= ~f
be the canonic series for A; here/is some integer, and the coefficients
An are pr-adic digits 0, 1,.. ., pr — 1. To the basis p these coefficients
can then be written as
r- x
An = 2j Unr + k P
k = 0
(n=-f, -/ + 1, -/ + 2,...), (2)
where the new coefficients aMf. + /c are p-adic digits 0,1,..., p — 1.
Hence A can be expressed as the p-adic number

54 The decomposition qfQ
Conversely, a p-adic number of the form (3) can again be written as a
pr-adic number by combining all terms ampm for which m lies between
nr and nr + r — 1, using the formulae (2).
By way of example,
2,1213423032... (5) = 2,78(22)(15)(17) .. .(25).
2 The ring Qg for integers g with the same prime factors
The result Qpr = Qp can be generalised. Denote by pl9p2, • • •, Pk
finitely many distinct primes, and put
9 = fi.-'tf? and 0' = Pi--.P*
where ru ... ,rk and r'l9..., r'k denote two distinct sets of k positive
integers. We assert that
% = Q.~
It is in fact not difficult to find two positive constants cl9 c2 such
that
cMg<\^\g'<C2\a\g
for all numbers in Q. Therefore, just as in the case k = 1, a sequence
{an} in Q is bounded, is fundamental, or is a null sequence, relative to
\a\ , if and only if it is the same relative to \a\ . From this the assertion
follows at once.
It is again possible to write every #-adic number as a #'-adic
number, and vice versa; but the change is a little less simple.
The identity Qg = Qg> shows in particular that it suffices to consider
only those g-adic rings Qg for which
g = p1p2-.-Pk (4)
is a product of distinct primes.
3 The components of a #-adic number
Assume that g has the form (4) where k is at least 2. Put further
9*=PlP2'-Pk-l-
The two completions Qg and Qg* are distinct. For a rational integer like
g*n is for large n divisible by a high power of #*, but is not even
divisible by the first power of g. Hence the sequence {#*, #*2, #*3,.. .}

The components of a g-adic number 55
is a null sequence relative to \a\g*9 but is not a null sequence relative to
1/71 whence the assertion.
\U\g9
Much more can be proved. Consider a fundamental sequence {an}
in Q relative to \a\g where g is as in (4). This means that, for all
sufficiently large suffixes m and n9 the numerator of am — an is divisible
by an arbitrarily high power of g9 while the denominator is prime to g.
This implies that the numerator of am — an is also divisible by
arbitrarily high powers of each of the primes pl9..., pk and that the
denominator is not divisible by any one of them. It follows at once
that the existence of the #-adic limit
9
A = lim an
n -*• oo
implies the existence of the k /?K-adic limits
Pk
lim an = AK9 say (k = 1, 2,..., k).
n -*■ oo
Conversely, the existence of all the limits AK evidently implies that
of A
Definition AK is called the pK-adic component of A, and one
writes
A = \A}, A2, • • • ? Ak/9
where the numbering of the components AK is the same as that of the
corresponding prime numbers pK.
The following property of the components is important.
The components AK of A do not depend on the special fundamental
sequence {an} by which A has been defined. (5)
Proof. If {a'n} is a second fundamental sequence such that
9
A = lim a'n9
n -*■ oo
then {an — a'n) is a #-adic null sequence. Hence, for all sufficiently large
?2, the numerator of an — a'n is divisible by an arbitrarily high power of
g9 while the denominator is prime to g. This implies again that the
numerator of an — an is divisible by arbitrarily high powers of each of
the primes pl9..., pk9 while the denominator is not divisible by any of

56 The decomposition ofQ
them. Hence {an — a'n} is a null sequence relative to each of the
valuations \a\ ,..., |a|Pk, and so the two sequences {an} and {a'n}
have for k = 1, 2,..., k the same pK-adic limit.
In particular, if the g-adic number A is defined as the sum of an
infinite series
00
a= Y,an (g),
n= 1
then the components AK of A are simply the series
00
Ak= £ an (PK) (k; = 1, 2,..., k).
n= 1
Assume in particular that the g-adic number g is given by its
canonic series
00
A= I anQn (#)'
n= -/
In order to find, say the component Ak of A, we write g = g*pk and
obtain the formula
00
n= -/
This is not yet the canonic /vadic series for ^4fc, but the reduction
method of Chapter 4 § 1 allows us to determine the canonic series.
The terms in negative powers of g may have to be treated separately;
but they are finite in number and so have a rational sum.
From the general rules for the sum, difference, and product of g-
adic and p-adic limits it follows that if B is a second #-adic number,
with the components
B = <#!, B2,..., Bk},
then
A + B = (Ax+Bl9A2 + B29...9Ak + Bk\
A-B = (A1-Bl9A2-B29...9Ak-Bky9
AB = (AlBl9A2B2,...,AkBky.
By way of example, let us again consider the two 6-adic numbers
A = 2,10101... (6) and B = 3,12022... (6)
constructed in Chapter 4 § 3 which had the property that AB = 0.

The numbers E{K) 57
Since
A = (AuA2y and B = {BUB2},
where Ax and B1 denote the 2-adic components, and A2 and B2 the
3-adic components of A and B, respectively, and since
0 = <0, 0>, and therefore iAtBl9 A2B2} = <0, 0>,
the components of A and B satisfy the equations
A1B1=0 and A2B2 = 0.
Since the components lie in fields, it follows that at least one of the two
2-adic numbers A x and Bu and at least one of the two 3-adic numbers
A2 and B2, are equal to 0. In fact,
,4 = 2 + lx6 + 0x62 + lx63+0x64 + lx65+... (6),
therefore
Ai = 2 + 3 x 2 + 0 x 22 + 27 x 23 + 0 x 24 + 243 x 25 +... (2),
and after reduction,
v41=0 + 0x2 + 0x22+0x23+0x24 + 0x25+... (2).
Similarly,
£ = 3 + lx6 + 2x62 + 0x63 + 2x64 + 2x65+... (6),
hence
£2 = 3 + 2x3 + 8x32 + 0x33 + 32x34 + 64x35+... (3),
and after reduction,
£2 = 0 + 0x3 + 0x32+0x33+0x34 + 0x35+... (3).
We see then that A1 and B2 are the vanishing components of A and B,
respectively.
4 The numbers E
(K)
The question now arises whether it is possible to choose the
components Al9 A2,..., Ak of a #-adic number A arbitrarily in the
respective fields Qpi, Qp2,..., Qpu. We shall prove that this is the case.
To begin with, let us construct for each suffix k = 1, 2,..., k a g-
adic number £(K) which has 1 as its Kth component, while the other
components are 0. For this purpose put
Pk = PiP2--PJPk (k = 1,2,..., k)

58 The decomposition ofQ
and
4K) = PnJ(pnK + K)
(k = 1, 2,..., k; n = 1, 2, 3,...).
Then
i-4k) = Pk/K + ^)-
By hypothesis the k primes pu ..., pk are all distinct. Therefore the
denominator p" + P" of 4° and 1 — 4° is divisible by none of these
primes and so is relatively prime to g = pr ... pk. It follows that, for k,
X=l 2,..., /c,
{4K)} is a /vadic null sequence for AJ=k, and
{1 — 4K)} is a pK-adic null sequence.
This means that all the /vadic limits
exist and have the values
PJL
lim
n -*• oo
■fi
:<^
if
if
, say,
k; = /I,
K=f= X.
\JC) A — 1, .Zi, ..., /CI
From this it further follows that also the g-adic limits
lim 4K) = E(K\ say, (k = 1, 2,..., k)
n -*■ oo
exist and that they have the components
E(K) = <Ku $K2, -•>, Kk> (k; = 1, 2,..., k).
Moreover,
/7(k) /7(A) _ )
(0 if k^,
and
£(D + £(2) + ...+ £(*) = < 1, 1, . . . , 1 > = 1.
5 The £-adic ring as a direct sum of pK -adic fields
As A = (Au ^42> • • • j A> runs °ver the #-adic ring Qg, each of its
components AK describes a certain subset Q* of the /?K-adic field. As
was proved by Hensel, Q* is in fact identical with Qp .

n as a direct sum of pK-adic fields 59
Theorem 1 Let AK,for k = 1, 2,..., /c, be an arbitrary pK-adic
number. Then there exists one and only one g-adic number A such
that
A = \Ai, A2, • • • , Ak/.
Proof. For each suffix k = 1, 2,..., k denote by {ajj0} a pK-adic
fundamental sequence of rational numbers such that
A„ = hm aiK).
k n
n -*• oo
There is of course no reason why for X =£ k the sequence {a^} should
also have p^-adic limits, and it may not even be bounded relative to
the valuation \a\ . We therefore apply the fact that the sequence {4K)}
has the pK-adic limit 1 and for X =£ k the p^-adic limit 0. It follows that
an infinite subsequence {e™} can be chosen such that
£„ fl(K)e<K> = K if k = X,
,,^ " r" [0 if K + X.
Hence
lira a<,K»e<Kn) = < 0,..., ^,,...,0),
n -» oo
with the component ^4K at the jcth place, and components 0 elsewhere.
Define finally a sequence {an} of rational numbers by
an = I aW (n = 1, 2, 3,...).
K= 1
As a sum of /c ^-adic fundamental sequences it is itself a fundamental
sequence, and it evidently has the g-adic limit
g k
lim an = £ <0,..., AK,..., 0> = (Au A2, • • ., Ak} = A.
n-+ oo k = 1
If a second g-adic number A' has the same components as A has,
then all the components of A — A' are equal to 0, and so A — A' is
the limit of a g-adic null sequence. This, however, requires that
A — A' = 0, as was to be proved.
In the language of algebra, Qg is the direct sum
Qg = QPl®QP2®-..®QPk
of the fields Qpi, Qp2,..., Qpk. It is for this reason that one usually
neglects the g-adic rings and deals only with p-adic fields.

60 The decomposition qfQ
6 Zero divisors and divisibility in Q
Let A, B9 and C be three g-adic numbers such that
A = BC9
and let
A = (Al9 A2,..-, Ak}, B = <£1? B2,..., £fc>,
C = <^ (^, G2, ..-, Cfc/
be their decompositions into pK-adic components. Then
Ai = B1C1, A2 = B2C2, • • •, Ak = BkCk,
where all the components lie in fields. Hence the following result
holds.
The equation BC = 0 is satisfied if and only if in each pair of
components
{Bl9 Ct}9 {B2, C2},..., {Bk, Ck}
at least one of the two elements is equal to 0. (6)
More exactly, let
Bp9 where p = pl9 pl9..., pr9
be all the vanishing components of B9 and let
Ba9 where o = al9 o29..., as9
be the components that are distinct from 0. The equation BC = 0
requires then that
Ca = 0, where o = ouo29... 9os9
but it imposes no restrictions on the remaining components
Cp9 where p = pl9 pl9..., pr.
Furthermore, under the same assumption for B, the quotient
A/B = C
exists if and only if
Ap = 0, where p = pl9 pl9..., pr9
while the remaining components
Aa9 where o = al9 ol9..., gs9
may be chosen arbitrarily in Qp. If r > 1 and these conditions are
satisfied, then the quotient C may be one of infinitely many #-adic

Numerical calculations 61
numbers; for its components
Cp, where p = pu p2,..., pr,
may be arbitrary pp-adic numbers, and only the remaining
components are fixed by
C<r = AJB<n Where 0- = 0-1,^---^5-
The result (6) shows that Qg always contains zero divisors when k > 2,
thus when g has at least two distinct prime divisors. From what has
been proved it further follows that in this case, for any given k, there
are always infinitely many #-adic numbers with the same /?K-adic
components.
7 Numerical calculations
There is another and more convenient way of finding the number E(K)
of § 4. The prime pK and the product PK as there defined are relatively
prime, and the same is true for their nth powers.
If now n g-adic digits of E(K) are wanted, determine a pair of integers
u and U such that
pnKu + PnKU = l.
The most convenient way of doing this is by means of the Euclidean
algorithm, as explained in elementary number theory. The number
e=l — pnKu
has now the properties
|e-l|PK<pK"n, and |e|w<£ft" for X±k.
Hence e — E(K) is divisible by gn, and e agrees with E(K) in the first n
digits.
By way of example, take g = 2 x 3, pl = 2, Pl = 3, and n = 7. The
equation
27w + 37L/=l
has the solution u = — 598 and U = 35. Therefore, in the conventional
notation to the basis 10,
e= 1 + 598 x 27 = 76545,
and on converting to 6-adic notation,
8 = 3,120531(6).

62 The decomposition ofQ
It follows that
£(1) = 3,120531 ... (6),
where this number has the 2-adic component 1 and the 3-adic
component 0. Since £(1) + £(2) = 1, we find further by subtraction
that
£(2) = 4,435024 ... (6)
has the 2-adic component 0 and the 3-adic component 1. Naturally
£<D + £<2) = 1, £(1)2 = £(D? £d)£(2) = 0, £(2)2 = £(2)?
and this can also easily be verified to seven digits by direct calculation
from the given values for £(1) and £(2).
By way of example, the 6-adic number with the 2-adic component 2
and the 3-adic component 3 is
2£(1> + 3£(2) = 3,535024 ... (6).
Generally, the 2-adic component of AE(1) and the 3-adic
component of AE(2) are the same as those of A itself. These components
can thus be obtained in 6-adic form by a simple multiplication, but
have still to be changed over into the 2-adic and 3-adic forms,
respectively.
There is yet another and more direct way of finding the #-adic
number A with given components. Let us say we want to find the first
digits of the 6-adic number
A = a^axa1a2i. .. (6)
which has the 2-adic component
At= -1 = 1,111 ...(2)
and the 3-adic component
^2=-1=1,111...(3).
This requires that
1 + 1x2 + 1x22 + 1x23 + ...
= a0 + ax x 6 + a2 x 62 + a3 x 63 + .. .
= a0-\-3a1 x 2 + 9a2 x 22 + 27a3 x 23 + ... (2),
and
1 + 1x3 + 1x32 + 1x33 + ...
= a0 + ax x 6 + a2 x 62 + a3 x 63 + ...
= a2 + 2a1 x 3 + 4a2 x 32 + 8a3 x 33 + ... (3).

Problems 63
Here apply the reduction process to the right-hand side series and
compare with the left-hand sides. It follows, first, that a0 — 1 is
divisible by both 2 and 3, hence that a0 is the 6-adic digit 1. Secondly,
3ax — 1 is divisible by 2 and 2al — 1 by 3, hence ax is the 6-adic digit 5.
Since 3ax — 1 = 7 x 2 and 2ax — 1 = 3 x 3, it follows next that 9a2 +
7 - 1 is divisible by 2 and 4a2 + 3 — 1 by 3, hence that a2 is the
6-adic digit a2 = 4. Continuing in this manner, we find that
A = 1,542015 ... (6).
8 Problems
1 Here and in the following problems let g = pl p2.. .pk, where k > 2,
be a product of distinct primes. Prove that every #-adic number has
a unique representation as a product of #-adic factors
A = A™A<2K..A<k\
where A(K\ for k = 1, 2,..., /c, has as its jcth component a certain
pK-adic number, while all its other components are 1.
2 Let g be as in Problem 1, and let A be the #-adic number
Find the polynomial
P(x) = xn + P1xn"1+... + Pn
with rational coefficients and of lowest degree n for which
P(^4) = 0.
Show that P(x) can be factorised into linear factors with rational
coefficients. Also determine the number of #-adic roots of the
equation
P(x) = 0.
3 Show that a #-adic number A is rational (i.e. lies in Q) if and only if
all its pK-adic components are equal to one and the same rational
number.
Prove that the digits in the canonic series of E(K) do not form a
periodic sequence.
4 Calculate the first digits of each of the three numbers E(1\ E(2\ and
£(3) when # = 2x3x5 = 30. Also determine all 30-adic numbers
which satisfy the equation X2 — X = 0.

64 The decomposition of Q
5 Find the first digits of the 10-adic number with the 2-adic
component ^ and the 5-adic component \.
6 Determine the first digits of the 2-adic and 3-adic components of
00 ^6
n = 0
7 The g-adic integer A has the components A = (Au A2,..., Aky.
Find the pK-adic component of
00
B= Y. nlA"-
n= 1
8 Let K = Q(x) be the field of rational functions in x with coefficients
in Q, and let P be a polynomial in x with coefficients in Q, at least of
degree 1. Show that to every element a =£ 0 of K there is a unique
integer f( > 0, < 0, or = 0) such that the numerator of Pfa is not
divisible by P and the denominator is prime to P. If
,, fO if a = 0
W(a)=\2f if a ±0,
show that w is a pseudo-valuation of K.
9 Let the hypothesis be as in problem (8), and let w0, wl9 and w2 be
the pseudo-valuations corresponding to the three polynomials
P0 = x(x — 1), Px=x, P2 = x—1.
Further denote by Kwo, KWl, and KW2 the completions of K relative
to w0, wl9 and w2, respectively. Show that KWi and KW2 are fields,
and that K„„ is the direct sum of these two fields.

6
The quadratic extension fields ofQp
The field Qp of all p-adic numbers is often called the field of rational p-
adic numbers because it is the completion of Q relative to the p-adic
valuation. In the present chapter we shall construct all the different
quadratic extensions of Qp that exist.
1 A distinction in the behaviour of the real field and the /?-adic
fields
A field K is said to be algebraically closed if every algebraic equation
xn + axxn ~ x + a2xn ~ 2 + ... + an = 0
with coefficients in K has one and therefore n roots in K.
It is proved in the algebra of fields that to every field K there is a
smallest algebraic extension field which is algebraically closed. This
extension is called the algebraic closure of K.
Consider in particular the real field R and the p-adic field Qp.
Neither is algebraically closed, for in R the equation
x2 + 1 = 0
and in Qp the equation
x2 - p = 0
have no roots. For the square of a real number is either 0 or positive
and so cannot be equal to — 1; and the square of the p-adic value of
every p-adic number is a square and so cannot be equal to p~ 1.
It was first proved by Gauss that the real field R has as its algebraic
closure the field C = R(i) of all complex numbers A = a + bi where
i2 = — 1, and a and b lie in R. On C, the valuation \a\ of R can be
continued into the valuation
|^|= +(a2+b2)112
of C which coincides with \a\ when A lies in R. Moreover, C is

66 The quadratic extension fields of Q
complete relative to \A\. In other words, there is only one single
extension field C of R.
By contrast, the p-adic field Qp has infinitely many distinct algebraic
extension fields, e.g. all the extension fields generated by roots of the
algebraic equations
xn - p = 0 (n = 2, 3, 4,...).
The algebraic closure of Qp, Qp say, is therefore of infinite degree over Qp.
One can continue the p-adic valuation \a\p into a valuation of Qp, but
then the further complication arises that Qp is not complete with
respect to this continued valuation.
We shall not deal with the general algebraic extensions of Qp, but
shall be satisfied with constructing only all the distinct quadratic
extension fields of Qp. As will be seen, there are only finitely many. Later
in Chapter 18 we shall investigate a class of functions defined on such
a quadratic extension field of Qp.
2 An equivalence relation
A quadratic extension field of Qp is obtained by adjoining to Qp a root
of some quadratic equation with coefficients in Qp. Without loss of
generality, this equation has the form
x2-d = 0, (1)
where d =/= 0 is a p-adic number which is not the square of a p-adic
number. As a consequence, the equation (1) cannot be solved in Qp
itself. On denoting a formal solution of (1) by y/d, the quadratic
extension field, Kp say, is then derived from Qp by adjoining ^/5,
Kp = Qp(y/d).
The elements of Kp can be written as
A = a + by/d, where a, beQp.
Two distinct elements d and d' of Q neither of which is 0 or the
square of a p-adic number, evidently produce the same extension field,
if and only if the quotient d/d' is the square of a p-adic number, a

Factors of d =^0 67
property which will be denoted by
d ~ d'.
Since evidently
d ~ d; if d ~ d\ then d' ~ d; if d ~ d' and d' ~ d", then d ~ d",
the symbol ~ defines an equivalence relation. We can therefore
subdivide the set of all non-zero elements of Qp into disjoint
equivalence classes by putting into the same class all numbers d ^= 0
that differ only by a square factor.
Our next problem will be to select in each such equivalence class
one representative.
3 Factors of d ± 0
Every p-adic number d =£ 0 has a unique representation
d = paS, where \d\p = p~a and 131,= 1. (2)
Here S is a p-adic unit with the canonic series
S = S0 + S'1p + S'2p2 + ...
where S0, S[, S'2, ... are digits 0, 1,..., p — 1, and where in particular
<50 =£ 0 and therefore \S0\p = 1.
Hence 3 can be written in the form
S = S0(l + S1p + S2p2+ ...), (3)
where also Su S2, • • • are digits 0,1,..., p — 1.
From now on denote by Sp the set of all p-adic numbers A satisfying
p/8 if P = 2,
The following definitions and facts from elementary number theory
will further be assumed.
If again g > 2 is a rational integer, two rational integers x and y are
called congruent modulo g, written x = y (mod g\ if |x — y\g < 1/g, i.e. if
g divides x — y. Further x is said to be a quadratic residue modulo g, or
a quadratic non-residue modulo g, according to whether x is, or is not,
congruent modulo g to the square of a rational integer.
If the rational integer y = 2n+l is odd, its square y2
= 4n(n +1)+1 is congruent to 1 modulo 8. Therefore of the four

68 The quadratic extension fields of Qp
odd numbers + 1, — 1, — 3, and + 3 the first is a quadratic residue
modulo 8, while the other three are quadratic non-residues modulo 8.
If further p > 3 is a prime, not all the integers prime to p are
quadratic residues modulo p, and hence there is a smallest positive
quadratic non-residue, N say. As can be shown, every integer not
divisible by p is now congruent modulo p to just one of the p — 1
integers
l2, 22,..., (i(p - l))2, l2Np, 22Np,..., (i(p - l))2Np.
Here the first \(p — 1) integers are quadratic residues, and the last
\{p — 1) integers are quadratic non-residues.
Consider now the formula (3) when p = 2. The factor S0 is then
equal to 1, and the second factor 1 + ^1x2 + ^2x22 + ... agrees in
its first three terms with exactly one of the four canonic series
+ I = l+0x2 + 0x22 + ..., -I = l + lx2+lx22 + ...,
-3 = l+0x2+lx22 + ..., +3 = l + lx2 + 0x22 + ...
which are 2-adic units. Therefore S has a unique representation
<5 = (-l)*(-3)cA,
where each of b and c is either 0 or 1, and where A = 1 (mod 8), hence A
lies in S2-
Thus we have proved that for p = 2 every 2-adic number d^O can in
just one way be written as
d = 2a(-lf(-3)cA, (4)
where a is some rational integer, each ofb and c is either 0 or 1, and A is
in S2.
Consider next the case when p > 3. Now the digit <50 has one of the
/7—1 possible values 1, 2,..., p — 1, and so there is a rational integer
x satisfying 1 < x < ^(p — 1) such that either
<50 = x2 (mod p\ or <50 = x2Np (mod p).
By putting b = 0 in the first and b = 1 in the second case, this implies
that
\S0 - Nbpx2\p < \jp and hence also |<50(A^x2)- x - \\p < l/p.
Therefore
S0 = Nbpx2A\ where A'eSp.
Since the second factor 1 + d1 p + S2p2 + ... in (3) also lies in S the

The elements ofSp are squares 69
same holds for the product (1 + S1p + S2p2 + .. .)A' = A, say. We
arrive therefore at the result that for p > 3 every p-adic number d =^ 0
can in a unique way be written in the form
d = paNbpx2A, (5)
where a is some rational integer, b is either 0 or 1, N is the smallest
quadratic non-residue modulo p, x is one of the integers 1, 2,...,
\(p — 1), and A is an element of the set Sp. This representation of d is
quite similar to the representation (4) in the 2-adic case.
4 The elements of SP are squares
The representations (4) and (5) of d will now be further simplified by
proving that the elements of Sp are squares of p-adic numbers. This
is done by means of the p-adic analogue of the binomial series for
(l+x)1/2.
We first assert that there exists a formal power series
f(x) = c0 + CiX + c2x2 + c3x3 + ..., where c0 = 1,
with rational coefficients such that its formal square
oo / m \
/(x)2 = l + 2Clx+ X X ckcm_k)xm
m=2\k=0 /
consists only of the first two terms,
f(x)2 = 1 + x.
This assertion evidently is satisfied if we choose
c -i
cl — 2
and define the coefficients ck with m > 2 by the recursive formulae
m
2j CkCm-k~v,
k = 0
that is,
m — 1
cm^-\Yu ckcm-k (m^2,3,4,...). (6)
k = 1
These formulae (6) show immediately that all the cm are rational
numbers. For p > 3, it is evident by induction on m that
\cm\p<l (m = l, 2, 3,...). (7)

70 The quadratic extension fields of Q
If, however, p = 2, only the weaker estimate
|cj2 ^2^-1 (m= 1,2,3,...) (8)
is true. For it certainly is satisfied if m = 1. If, however, m > 2 and the
estimate has already been proved for all suffixes up to m — 1, then it
also holds for the suffix m because, by (6),
|cj2<2x max 22*"1 x 22{m~k)- x =22m~ K
1 < k < m - 1
We are interested mainly in the partial sums
fn(x) = 1 + cxx + c2x2 + ... + cnxn (n = 1, 2, 3,...)
of/(x). Their squares have the form
fn(x)2 = l + x+ £ Cmnxm (9)
m = n + 1
where, by (6), the terms in x2, x3,..., xn are missing and the new
coefficients Cmn are for n + 1 < m < 2n given by
mn
n
^mn ~ 2j
CuC
k^m - k'
k = m — n
Hence, by (7) and (8),
\Cmn\2< max 2^-^2^-^-^22^2 if p = 2;
m — n < k < n
|CmJp<l if p>3. (10)
Next denote by x an arbitrary p-adic number such that
1/8 if p = 2,
X]p- ' 1/p if p>3.
In other words, 1 +x lies in Sp. By (7) and (8), as m-> oo,
0<|cmxm|2<2|2m"1)_3m = 2-m-1 ->■() if p = 2,
and
0<|cmxm|p</Tm^0 if p>3.
Therefore the limit
p oo
lim/n(x)=l+ £ <^n*n =/(*)> say,
n-* oo n = 1
exists for both p = 2 and p > 3. Next, by (10), as n -> oo,
In
c
mn
0<
/ ^ ^J*!^-^
m = n + 1
< max 22m-2x2~3m<2-n-3->0
2n+l<m<2n
if p = 2,

The distinct quadratic extensions ofQp 71
and
0<
In
c
mn
2j ^r"*X
< max lxp"m<p"""UO if p>3,
n + 1 < m < 2n
m = n + 1
so that, by (9), again for both p = 2 and p > 3,
f(x)2 = 1 + x.
Finally apply this result to x = A — 1 where A is any element of Sp
This leads to the relation
(/(A - 1))2 = A,
and thus gives the following lemma.
Main Lemma Every element ofSp is the square of a p-adic number.
5 Enumeration of all distinct quadratic extensions of Qp
The formulae (4) and (5) and the Main Lemma allow us now to make
the following statements.
Every 2-adic number d / 0 has a unique representation
d = 2a(-l)\-3)cX2 (11)
where each of the exponents a, fe, and c is either 0 or 1, and where
X / 0 is a p-adic number.
For p>3 every p-adic number d^O can in just one way be written
as
d = paNbpX2 (12)
where Np is the smallest quadratic non-residue modulo p, each of the
exponents a and b is either 0 or 1, and X / 0 is a p-adic number.
In the 2-adic case d is a square only when a = b = c = 0, and in the p-
adic case when p > 3 it is a square only if a = b = 0. For if/? = 2, then d
certainly is not a square if a = 1; and if a = 0, but at least one of b and c
is not 0, then the factor
(-l)6(-3f
is not congruent to 1 modulo 8 and therefore is not a square. Further,
if p > 3, then d is not a square if a = 1; and if a = 0, but b = 1, then
again Nbp is not congruent to a square modulo p and so is not a square.
In the equivalence notation of § 2 there are for p = 2 exactly 8

72 The quadratic extension fields of Qp
distinct equivalence classes which correspond to the 8 distinct triplets
(a, ft, c) where each of a, ft, and c is either 0 or 1. If further p > 3, then
there are 4 equivalence classes corresponding to the pairs (a, ft) where
again each number a or ft is either 0 or 1. We need only consider the
equivalence classes that do not consist of squares. To each of the
remaining equivalence classes we obtain then 7 corresponding
distinct quadratic extensions of Q2, and for p > 3 there are 3 distinct
extensions of Qp, as follows.
Theorem 1 There are exactly 7 distinct quadratic extensions of
Q2, and these may be represented by
22(7^6), Q2(^6) (13)
If p > 3, then Qp has exactly 3 distinct quadratic extensions, and
these may be represented by
QPk/N~P), QP{-Jp), QPkfWP\ (14)
By way of example, 22(v^) is tne same extension field as 22(V' ~ 3),
and 22(\/T7) is identical with Q2 itself; for — 3 ~ 5 and 1 ~ 17 because
-=3^- and 17 lie in S2. Further, if p > 3 and d ± 0 is not divisible by /?,
then Qp(s/d) coincides with either Qp or with Qpiy/Np), depending on
whether d is a quadratic residue or non-residue modulo p.
6 The arithmetic in the quadratic extension fields of Qp
For both p = 2 and /7 > 3 denote by Xp = 2^(^/¾ any one of the 7 + 3
quadratic extension fields (13) or (14). On putting formally
every element z of Kp can be written in the form
z = x + yj,
where x and y are in Qp. In the abstract definition of the quadratic
extension Kp one forms the ring Qp[f\ of polynomials in an
indeterminate t with coefficients in Q but identifies any two
polynomials when their difference is divisible by t2 — d. Then j is the

Continuation of \x _ to Kn 73
class of all polynomials that differ from i by a multiple of the
polynomial t2 — d.
It is easy to show that if
z' = x' + y'j
is a second element of Kp9 sum, difference, product, and quotient of z
and z' are given by the equations
z + z' = (x + yj) + (x' + y'j) = (x + x') + (y + y')j9
z-z' = (x + yj)-(x' + y'j) = (x-x') + (y-y')j,
zz' = (x + yj)(x' + y'j) = (xx' + dyy') + (xy' + yx')j9
— - x' + y'J _ (**' - dyy') + (xy' - yx'),/
z x + yj x2 — dy2
Here division is possible exactly when z =£ 0, thus when at most one of
the p-adic numbers x and y is 0, for then x2 — dy2 ^= 0 since d is not a
square.
These four formulae for z + z\z — z', zz', and z'/z are analogous to
those in the complex number field; note, however, that x, y, x', and y'
are now p-adic and not real numbers!
Also in analogy to the complex field, the conjugate z of z = x + yj is
defined by
z = x- yj,
and the trace S(z) and the norm N(z) by
S(z) = z + z = 2x and Af(z) = zz = x2 — dy2.
Both S(z) and AT(z) lie in Qp9 and they are connected by the quadratic
equation
Z2 - S(z)Z + N(z) = 0 (15)
which has the two roots z and z.
Evidently for all z and z' in Xp,
S(z + z') = 5(z) + S(z')9 N(zz') = N(z)N(z').
7 The continuation of the/i-adic valuation | x\p to iS^
The p-adic field Q 'has the non-archimedean valuation IxL. It is a
general theorem that this valuation can be continued to every
algebraic extension of Qp. We shall now show, as a special case, that

74 The quadratic extension fields of Q
\x\p can be continued to a valuation of Kp, and that then Kp is
complete with respect to this continuation.
The proof is based on the following result.
Theorem 2 If the number z = x + yj of Kp satisfies the two
inequalities
\S(z)\p^p, |AT(z)|p<l,
then z lies in Qp.
Proof The quadratic equation (15) has the roots z and z given
explicitly by
${S(z)±{S(zf-4N(z))1'2}.
They lie in Qp if and only if S(z)2 — 4N(z) is the square of a p-adic
number. This is the case exactly when the expression
A = 1 - 4N(z)/S(z)2
is a square. But, by the hypothesis,
|A-1|2<2"4 if p = 2, and \A-l\p<p~2 if p>3.
Hence in both cases A belongs to the set Sp and hence, by the Main
Lemma, is the square of a p-adic number.
For every element z = x + yj of Kp put now
w(z) = \N(z)\1p'2 = \x2-dy2\1J2,
where the square root is taken with the positive sign. If z lies in Qp9 so
that y = 0, then
w(z) = \x\p
coincides with the original p-adic valuation of Qp. We show now that
w(z) is a non-archimedean valuation on Kp.
Since N(z) vanishes only when z = 0, it is clear that
(i) w(0) = 0, but w(z)>0 if z^O.
Secondly, the equation N(zz') = N(z)N(z') implies that
(ii) w(zz') = w(z)w(z').
Thirdly, we show that w satisfies the triangle inequality in the non-
archimedean form,
(iii) w(z + z')< max (w(z), w(z')).

The Kp-integers 75
For this purpose it may be assumed that z ^ 0 and, say
w(z) < w(z').
On putting
Z = zjz'
and using the property (ii), the assertion (iii) takes the equivalent form,
If w(Z)<l, then also w(Z+1)<1.
This property certainly is true when Z lies in Qp. Let therefore Z be
in Kp, but not in Qp. The hypothesis w(Z) < 1 is equivalent to
\N(Z)\p < 1.
Since Z is not an element of Qp, Theorem 2 shows that also
\S(Z)\P<1.
Hence
\N(Z + 1)1, = |(Z+ 1)(2+ 1)1, = |ZZ + Z + 2+II,
= |iV(Z) + S(Z)+l|p<l,
implying (iii).
The three properties (i), (ii), and (iii) together show that w(z) is a
non-archimedean valuation on Kp9 and we have already seen that on
Qp it is identical with |x| . This is what we mean by saying that w(Z) is
a continuation of \x\n.
It can be proved that |x| has no other continuations on Kp, but
since this property will not be needed, the proof will be omitted.
In future, we shall write \z\ for w(z) even when z belongs to Kp.
8 The i^-integers
Denote by IK the set of all elements z of Kp for which
1*1,^1.
Such elements of Kp will be called Kp-integers. It is obvious from what
has been proved that the set Ip of all p-adic integers is a subset of IK.
It is possible to give a simple characterisation of the X,-integers
z = x + yj in terms of properties of x and y. This characterisation will,
however, be slightly different for the one extension field
Q2(V^) = Kt say,

76 The quadratic extension fields of Q
compared with the other extensions (13) and (14).
Theorem 3 Let z — x + yj be a Kp-integer. Then
max(|x|p, \y\p)<l if Kp*K%,
max(131,,13^2 if Kp = K%.
Proof Since z is a j^-integer, necessarily
\z\2p = \x2-dy2\p<\. (16)
Excluding the trivial case when z = 0, there is an integer f such that
max(\x\p9\y\p) = pf. (17)
If / < 0, the assertion is true; let therefore, without loss of generality,
The 7 + 3 fields (13) and (14) have the property that d is divisible by
at most the first power of p, so that \dy2\p>p~1\y\p and therefore, by
(17),
max (I*2!,, \dy2\p)>p2f~l>p.
It follows then from (16) that
\AP = \dy2\r (18)
This equation next shows that d cannot be divisible by p because
otherwise the left-hand side would be an even power and the right-
hand side an odd power of p. Hence Kp must be one of the four fields
Q2(V^)> QiiV^, QziVl) where p = 29
and Qpiy/Np) where /?>3,
for which d is not divisible by p, and (18) takes the simpler form
i*2ip = iA, (2°)
and so, by (17),
\x\p = \y\p = pf-
Put now
A = dy2/x2.
Since f > 1, it follows from (16) that
|1 -A|p = \x-% = p-2'<p-2. (21)
This allows us already to exclude the possibility that p = 2 and / > 2
or that p > 3 and f > 1; for in either case it would follow from the

The field Kp is complete 77
Main Lemma that A and hence also d were squares in Qp9 which is
false.
That leaves only the case when p = 2, and Kp is one of the first three
fields (19). Here the two fields Q2(J^i) and Q2{~J?>) can also be
excluded. For the quotient y2/x2 is a square with the 2-adic value 1,
hence, as is easily shown, has a canonic series with the first terms
1 + 0x2 + 0x22 + ...
The assumption that d= — 1 or d = 3 leads therefore to the equation
|l-A|2=i
contrary to (21).
The last remaining field Q2{\f — 3) = i£* does in fact contain
numbers z with f = 1, as the example of the Xf-integer
z = i(l+y^3) with AT(z) = (A)2-(-3)(A)2 = l
shows. This concludes the proof.
9 The field Kp is complete with respect to its valuation
Whenever the element z = x + yj of Kp does not vanish, there is a
unique rational integer g such that
f~'<\z\p<f,
hence also
\P9AP < 1.
Hence p9z is a j^-integer, and so by Theorem 3
maxflxl,, \y\p)<2p* <2p\z\p. (22)
This estimate allows to prove the following theorem.
Theorem 4 Every quadratic extension field Kp ofQp is complete
with respect to its valuation \z\p.
Proof. Let {zn}, where zn = xn + yj with xn and yn in Qp9 be an
arbitrary fundamental sequence in Kp with respect to \z\p. This means
that to every positive number e there is a positive number g(e) such
that
\zm — zn\p<s if m>q(s) and n>q(s).

78 The quadratic extension fields of Qp
By (22), this further implies that also
max(\xm-xn\p9\ym-yn\p)<2pe if m>q(s) and n>q(e);
for xm = xn and ym = yn if zm = zn. Hence both {xn} and {yn} are
fundamental sequences in Qp and so have certain p-adic limits
p v
x = lim xn and y = lim yn.
n -*■ oo n -» oo
Denote by z the element z = x + jtf in j£p. Then, as n tends to infinity,
0 < \zn - z\p < max(|xn - x\p9 \yn - y\p\j\p)^>0.
Hence {zn — z) is a null sequence, and therefore {zn} is a fundamental
sequence, relative to the valuation \z\ in Xp, and this fundamental
sequence has the limit z. This proves the assertion.
10 A basis representation of the Kp -integers
The set IK of all Xp-integers was defined by the inequality
\z\2p = \x2-dy\<l.
Therefore z certainly is a j^-integer if x and y are p-adic integers; and
if Kp is not the special field K*9 then, by Theorem 3, this sufficient
condition is also necessary.
If, however, Kp is the special field j£f, then put
r = i(_l+x/33).
This number is a Xf-integer since its norm has the 2-adic value
U*l2 = l(-i)2-(-Wl2 = i.
Every element of Xf can now be written in the form
z = x* + y*j* = (X* - $y*) + i^V^I, (23)
where x* and y* are suitable 2-adic numbers. In this notation,
1*1! = l(** - h*)2 - (~ 3)(iy*)2|2 = I**2 - x*y* + y*2\2.
Hence z is a Xf-integer if and only if
|x*2 — x*y* + y*2|2 < 1.
We assert that this inequality is satisfied if and only if both x* and y*
are 2-adic integers. That this condition is sufficient is clear; it only
remains to be proved that it is also necessary.

Ramified and unramified fields Kp 79
If we assume that z is a Xf-integer, it follows from Theorem 3 and
from (23) that
max(|x*-^*|2, \?y*\2)<2.
It follows immediately that
|y*l2<l,
hence that
either |x*|2<l or |x*U = 2.
The second possibility can be excluded because it would imply that
|x*2 — x*y* + y*2|2 = \x*2\i > 1j
against the hypothesis that z is a Xf-integer.
Hence, with a slight change of notation, the following assertion has
been proved.
Theorem 5 Let Kp = Qp (y/d) be one of the 7 + 3 distinct
quadratic extensions (13) and (14) of Q 9 and let further
. f Vd if Kp + K-t,
J U(-1+7^3) if Kp = Kl
Then z = x + yj is a Kp-integer if and only if x and y are p-adic
integers.
11 Ramified and unramified fields Kp
Let/ be as in Theorem 5, and let z = x + yj be a Kp-integer. Since x and
y are p-adic integers, they can be written as canonic series
x = x0-\-x1p-\-x2p2 + ...(/?) and y = y0 + ytp + y2p2 + .. . (/?),
where the coefficients xn and yn are digits 0,1,..., p — 1. This gives for
z itself a development
z = (x0 + yj) + (x1 + yj)p + (x2 + y2j)p2 + ...(/?)
which evidently is unique.
This development can be simplified, but in two different ways,
depending on whether Kp is one of the 6 + 2 fields
Q2(J^X QiUfi), 62(73¾ Qiiyfh Qi(J^>\ e2(%/6);
6p(>/p), QP{y/WP\

80
The quadratic extension fields of Q
P =
or is one of the 1 + 1 fields
(U) QiU^Y QPUN~Py
Here p > 3.
First let Kp be one of the fields (R). Put
1+j if Kp = Q2(^A) or Kp = Q2(
j for the other 6 fields (R).
A simple calculation shows that for all 8 fields,
IPI =P"1/2
\a \p f ?
and that p can in Kp be factorised in the form
p = EP\
where E is given in the following table,
3)
(24)
E= -
E = l
E= -1
£ = 2-
£=+i
£=1
£ = 1/JV,
1 = 1-P
3 = 3-P
if
if
if
if
f
f
f
f
Kp = Q2(y/2\
KP = Q2(^2\
Kp = Q2(y/3\
Kp = Qp(y/P\
Kp = QJ^/^A
P = 2,
P = 2>
V = X
P = 2,
P = 2,
P>\
P>3.
s equal to 1. Hence not only £, but
In each case the p-adic value of £
also E" 1 is a Kp-integer.
In the fields Kp the prime p becomes essentially a square; for this
reason the fields (R) are called ramified. It is easily verified that for each
such field both numbers j and £ are linear polynomials in P with
integral p-adic coefficients. The earlier development of z can therefore
be replaced by one of the form
z = z0 + z1P + z2P2 + ...(F), (25)
where the coefficients zn still are digits 0, 1,..., p — 1, but where p has
now been replaced by P. This series (25) holds for the Xp-integers. For
general elements of Kp we have to add finitely many similar terms in
negative powers of P.
Secondly, let Kp be one of the two fields (U), and let j be as in
Theorem 5. If z = x + yj is a Xp-integer, then by this theorem both x

Problems 81
and y are p-adic integers. The norm of z is
N(z) = x2 - xy + y2 if Kp = Q2(y/^3\
and
N(z) = x2 - Npy2 if Kp = Qp(y/N~p),
and it is not divisible by any odd power of 2 or of p, respectively. It
follows that, unlike the case of ramified fields, there exists no
Kp-intQgQr P satisfying
IPI =v~112
and therefore p cannot be factorised in the ring ofKp-integers. It is for
this reason that the two fields (U) are said to be unramified. For each of
the fields Q2, and Qp where p > 3, we have thus found exactly one
unramified quadratic extension. This is a special case of a general
theorem on algebraic extensions of Qp.
The general i^-integer z in such an unramified field has the
expansion
Z = ZQ+ ZyP + Z2P2 + . . .
in powers of p where, however, the new digits Zn have the form
r + sj, where r, s = 0,1,..., p — 1.
If z is not a j^-integer, then finitely many terms in negative powers of
p must be added, but these have as coefficients digits of the same form.
In the last chapter of this book, we shall apply the preceding results
to the study of functions on IK with values in Kp. Also there the
distinction into ramified and unramified extensions will play an
important role.
12 Problems
1 Determine with which of the fields (13) or (14) or Qp the following
quadratic extensions are identical.
e5(V-iooi),e5(yiooo).
2 Decide whether the fourth root>7 — 1 lies in Q2 or in one of its seven
quadratic extensions.
3 Decide whether ^/- 1 lies in Q5 or in one of the three quadratic
extensions of this field.

82 The quadratic extension fields of Q
4 A formal power series
00
/=1+ X dkx>
satisfies the formal identity
/3 = l+x.
Show that the coefficients dk are rational numbers, give recursive
formulae for them, and derive upper estimates for
\dk\p (£=1,2,3,...),
when (i) p = 2, (ii) p = 3, and (iii) p > 5.
5 If dk is as in Problem 4, study the coefficients Dmn in
1 + £ dk*k )= 1 + * + Z ^--^
3n
fc=l / m=n+l
and establish upper estimates for
\Dmn\P (n + 1 < m < 3n; n = 1, 2, 3,...),
when (i) /? = 2, (ii) p = 3, and (iii) /? > 5.
6 Apply the results of the last questions to determine an integer t
(which may depend on p) such that if A is a p-adic number satisfying
|A-l|,<p-f,
then A is the cube of a p-adic number.

II. FUNCTIONS

7
Elementary topological properties of
/?, Qp, and Qg
Now that we have dealt with p-adic and g-adic numbers and their
arithmetic in the first part of this book, the second part will treat of
functions of such numbers. It will be concerned mainly with p-adic
valued functions of one variable on the set
IP- \x\p£l
of p-adic integers, or on the set
J={0, 1,2,...}
of all non-negative rational integers. But since a great deal of the
theory carries over with little change to this more general case, we
shall in the beginning also consider #-adic valued functions on the set
of g-adic integers, or on the set J.
We shall try in particular to learn to understand the similarities and
the differences of this kind of function as compared with that of real-
valued functions on an interval
a < x <b
in the real field, as discussed in real analysis, and we shall compare the
properties of continuity and differentiability of p-adic functions with
those of real functions.
1 Rings with a pseudo-valuation as metric spaces
The similarities and differences in the behaviour of such functions can
be understood best in terms of the elementary topology of the real
field on one hand, and of the p-adic field and the #-adic ring on the
other hand. These fields and rings can be interpreted as metric spaces,
owing to the distances derived from their valuation or peudo-
valuation.

86 Elementary topological properties of R, Qp, and Qg
Any set S of arbitrary elements a,b,c,... becomes a metric space
(S, d) if a real valued function d(a, b) of any two (equal or distinct)
elements a.boiS is given which has the following properties.
(i) d(a, a) = 0, d(a, b)>0 if a =£ b.
(ii) d(b,a) = d(a,b).
(iii) d(a, c) < d(a, b) + d{b, c).
We call d(a, b) the distance from a to b. If E is any subset of S, d is
defined also on £, and hence the pair (£, d) is a metric subspace of(S, d).
In this book we are interested mainly in those metric spaces that
can be derived from rings K with a pseudo-valuation (or valuation)
w(a). By Chapter 2, such pseudo-valuations have the following
properties.
w(0) = 0, w(a)>0 if a ^0.
w( —a) = w(a).
w(a + ft) < w(a) + w(fe).
Hence, on putting
d(a, b) = w(a — b\
d satisfies the properties (i), (ii), (iii) of a distance, and so (K, d) is a
metric space. The same is true for (E, d) where E is any subset of K.
In the opposite direction, (X, d) can possibly be enlarged. Let Kw
be the completion of K relative to w, and w(A) the continuation of w(a)
on K to the elements A ofKw. Then w(A) is a pseudo-valuation on Kw,
and hence (Xw, d) where
d(A, B) = w(A - B) (A, B in KJ
is again a metric space; it evidently contains (X, d) as a subspace.
We found in Chapter 2 that to every e > 0 and every ^4 in Xw there is
an a in K such that
w(^4 — a) < a, thus d(^4, a) < e.
In other words, the elements of (X, d) lie everywhere dense in (Kw, d).
2 Notations and definitions
The following notations will be used.
The symbols e, ¢, c=, u, n have their usual meaning from set
theory; however, the sign c= does not exclude identity.

Notations and definitions 87
The elements of a metric space are also called its points; this name
of point will in particular often be used for the elements of a set of real,
p-adic, or g-adic numbers. The letter £, with or without suffixes, will
denote subsets of the whole set K, and 0 will be the empty set.
A collection of sets is called disjoint if no two of its sets have a point
in common.
If a is any point of (K, d) and p is a real positive number, the ball
U(a, p) is the set of all points of (K, d) satisfying
w(x — a) < p.
We call a the centre and p the radius of U(a, p).
It follows easily from the property (iii) of d that if b e U(a, p\ and if
a = p — w(a — b\ then
the ball U(b, a) is a subset of U(a, p).
The set E is said to be bounded if it is contained in some ball U(a, p)
and therefore is a subset of the larger ball £/(0, p + w(a)) with centre
atO.
If E is bounded, then every sequence {an} of points of E is
w-bounded.
The complement %?E of E (relative to K) consists of all points of K
which are not in E. Hence
Ev%E = K and En<£E = 0.
The point a of K is a point of closure ofE if for every a > 0 the ball
U(a, a) contains at least one point x of E.
The point a of K is an interior point ofE if there exists some positive
number a such that the ball U(a, a) is a subset of E.
The closure E of E is the set of all points of closure of E, and the
interior JE of E is the set of all interior points of E. Here
EczE, E=E, and JEaE, J(JE) = JE.
The set E is said to be closed if E = £, and it is said to be open if
E = J>E. If E is an arbitrary subset of K, then E is always closed, and
JE is always open. All balls V(a, p) are open sets, and as we shall see,
in the case of (Qg, d) they are also closed, but not in that of (R, d).
The frontier 3FE is defined by
^E = En^E
and is always a closed set.

88 Elementary topological properties of R, Qp, and Qg
A set £ is dense in K if E = K. By way of example, K is dense in Kw.
The set E is said to be (sequentially) compact if to every infinite
sequence {an} = {a1? a2, a3,...} of points of £ there exists an infinite
subsequence
{ari, ar2, ar3,...}, where 1 < r1 < r2 < r3 < ...,
and a point a of £ such that
lim w(a — ar ) = 0,
n -*■ oo
This means in the notation of Chapter 2 that
w-lima,. =a.
n -*■ oo
rn
A collection of finitely many closed subsets £1? £2,..., En of £,
where n > 2, is said to form a partition of E if
(i) any two of these subsets are disjoint, and
(ii) E=\J Ek.
k= 1
If E has no such partition, then it is called connected.
The meaning of these definitions for the balls in £, Qp, and Qg will
soon be studied. But we first state without proof a number of well-
known properties of closed or open sets.
The two complementary sets K and 0 are both open and closed.
IfE is open, then %?E is closed, and ifE is closed, then %?E is open.
The union of finitely many, and the intersection of any collection of,
closed sets is closed. On the other hand, the intersection of finitely
many, and the union of any collection of, open sets is open.
3 The balls as finite intervals in R
As a first example take for K the real field R with the valuation
w(a) = \a\, so that the distance becomes d(a, b) = \a — b\. Now the ball
U(a, p) is the open interval
\x — a\<p or a — p < x <a + p
on the real line, and its closure U(a, p) becomes the closed interval
\x — a\<p or a — p <x <a + p.
The frontiers of both U(a, p) and U(a, p) consist of the two points
a — p and a + p.

The balls in Qg 89
The general open and closed intervals
a < x < b and a<x <b
are identical with the ball U(\(a + b\ \(b—a)) and its closure
U{\{a + b\ \(b—a)\ respectively. Thus finite intervals are not both
open and closed, and the two half-open intervals
a<x <b and a <x<b
are neither open nor closed.
It can be proved that finite intervals are connected.
4 The balls in Qg
Next let K be the #-adic ring Qg with the pseudo-valuation w(a) = \a\ ,
hence with the distance d(a, b) = \a — b\ . The only values assumed by
the g-adic pseudo-valuation are 0 and rational integral powers of g,
and this pseudo-valuation is non-archimedean. As a consequence the
balls in (Qg, d) have properties quite different from those in (R, d).
It is clear that we need only consider balls U(a, p) where p is an
integral power of g, say p = g~s+1 where s denotes a rational integer.
This ball will in future be denoted by U(a; s); it consists of all #-adic
numbers x satisfying
\x—a\g< g~s+1, or equivalently, \x—a\g<g~s.
The following lemmas describe a number of properties of such balls.
Lemma 1 IfbeU(a; s), then U(b; s) = U(a; s). In other words,
every point of U(a; s) is a centre of this ball.
Proof. Let x be any point of U(b; s). Then by the hypothesis,
\a-b\g<g~s and \b-x\g<g~\
and therefore, by the non-archimedean triangle inequality,
\a -x\g = \(a-b) + (b- x)\g < max (\a - b\g, \b - x\g) <g~s.
Hence every point x of U(b; s) lies in U(a; s), whence U(b; s)czU(a; s).
Now the condition
\a — b\n<q~s
for b to lie in V(a\ s) is identical with that for a to lie in U(b; s). It

90 Elementary topological properties of R, Qp, and Qg
follows that also U(a;s)czU(b; s), proving the identity of the two
balls.
Lemma 2 The set U(a\s) is both open and closed.
Proof. From the definition of open sets it follows at once that U(a; s)
is open. In order to prove that V(a\s) is also closed, it suffices to show
that the complementary set ^U(a; s) is open. Let b be any point of
^U(a;s); hence b does not belong to U(a; s). Assume there exists a
point c of U(b; s) which does not lie in ^U(a; s) and so lies in U(a; s).
Then, by Lemma 1, both U{c; s) = U(a; s) and U(c; s) = U(b; s), and
therefore U(a;s)=U(b; s). This implies that b lies in U(a; s), contrary
to the hypothesis. We conclude then that no such point c can exist,
hence that U(b;s) is contained in (€V(a\s) and that therefore
^U(a; s) is an open and U(a;s) a closed set.
Lemma 3 The frontier &rU(a\s)ofU(a\s) is the empty set (/)
Proof. ^V(a\s) is the intersection of the closures of V(a\s) and
^U(a; s). But these two sets are both closed and disjoint, giving the
assertion.
Lemma 4 The ball U(a; s) is not connected.
Proof. We must show that there exists a partition of U(a; s). Now
every point b of V(a\ s) can be written in the form
b = a + gsx,
where x is a certain #-adic integer. Therefore x can be put in the form
x = x0 + gy,
where x0 is one of the digits 0, 1,..., g — 1, and y is again a #-adic
integer. Now
b = (a + gsx0) + gs+1y.
If in this formula x0 is a fixed digit, and y runs over all #-adic integers,
then b runs over all points of the ball U(a + gsx0; s + 1). Since two
such balls belonging to different digits x0 evidently are disjoint, it

Two properties of compact sets 91
follows that U(a; s) is the union
0-1
U(a;s)= [j U(a + gsx0\s+l).
*o = 0
This is a partition of U(a; s) into g > 2 disjoint closed sets and shows
that U(a; s) is not connected.
The set of all balls U(a, p) in R is not countable. For, as Cantor
proved, the set of positive real numbers is not countable. The same is
therefore true of the set of centres a, and the set of radii p, and so even
more of the set of all balls U(a, p).
A completely different result holds for the set of all balls U(a;s) in
Qr
Lemma 5 The set of all balls U(a; s) in Qg is countable.
Proof Write the centre a of [/(a; s) as a canonic series
00
a= Z anGH
n= ~f
and put
s- 1
A= Z anQ\
n= ~f
so that A is a rational number. Evidently A e V(a\ s) and therefore, by
Lemma 1,
U(A;s)=U(a;s).
Here both the centres A and the integers s form countable sets. The
same is then true for the product set of all pairs (A, s) and so also of the
set of all balls U(A; s).
5 Two properties of compact sets
For the moment let K be again any ring with a pseudo-valuation w(a),
and let (K; d) be the metric space with the distance d(a, b) = w(a — b).
Further let £ be a compact subset of K.
Lemma 6 The compact set E is bounded.
Proof. If the set E is not bounded, then an infinite sequence {an} of
points of E can be selected as follows.

92 Elementary topological properties of R, Qp, and Q
The point al is arbitrary in E\ the point a2eE is chosen such that
w(a2-a1)> 1,
and from here on the points are chosen in E such that
w{an + 1—a1)> w(an — ax) + 1 (n = 2, 3, 4,.. .).
Then
w(am — a^ > w(an — ax) + 1 if m> n,
hence
w(flm ~an) = w((am -ax) - (an -at)) > |w(am -ax) - w(a„ -^)1 > 1.
This inequality shows that no subsequence of {an} is a fundamental
sequence and that the hypothesis is therefore false, hence that E is
bounded.
Lemma 1 Every compact set is closed.
Proof Denote by a an arbitrary point of closure of E. There exists
then an infinite sequence of points {an} of E with the w-limit a, and
every infinite subsequence {arJ of {an} has the same w-limit a. The
definition of compact sets implies then that a belongs to E9 and this
shows that E is closed.
In the special cases of balls U(a, p) in R and U(a; s) in Qg we shall
prove that the closed and bounded sets t/(a, p) and U(a; s) are also
compact. This is in fact true for all closed and bounded sets in these
spaces, but will not be used.
6 The closed balls in R are compact
In the study of continuous functions in the real field R, with the
distance d(a, b) = \a — b\, a basic property used states that the closed
interval
i :a<x < ft,
or equivalently, the closed ball U(\(a + ft), \(b —a)\ is a compact set.
Let us sketch a proof of this property.
Choose an integer f( > 0, < 0, or = 0) such that
max(\a\,\b\)<10-f,

The balls in Qg are compact 93
and take an arbitrary infinite sequence {an} of points in i. Each element
a of this sequence can in at least one way be written as an infinite
decimal fraction
00
a„ = enZ am„lQ-f-m+1 (n= 1,2,3,...),
m = 1
where each sn is either + 1 or — 1, and where the coefficients amn are
decadic digits 0,1,..., 9. We call sn the sign, andamn the mth digit ofan.
Here it is important that each of the numbers sn and amn has only
finitely many possible values.
To prove the compactness of i, it suffices to find a convergent
subsequence of {an}; for since i is a closed set, the limit lies then
also in i.
For this purpose it is now possible to select in {an} an infinite
sequence of consecutive subsequences
Zk = {<*?} (k = 0, 1, 2,...)
with the following properties,
(i) Z0 is a subsequence of {an}, and Ik is a subsequence of Zk _ x for
k>l.
(ii) The signs of all the terms of I0 and hence also the signs of all the
terms of every following sequence Ik have the same value, e say.
(iii) For every suffix k the /cth digit of every term of Ik9 and hence also
the /cth digit of every term of II where I > k, have the same value,
a{k) say.
Having in this manner chosen the sequences Ik9 we select a final
subsequence
of {an}. It is clear from the construction that this sequence has the limit
00
e ^ 0(-)10-/--+1.
m = 1
and since i is a closed set, it contains this limit. Hence I is compact.
7 The balls in Qg are compact
We gave the last proof because an almost identical construction
allows us to show that also every ball U(a;s) in Q is compact.

94 Elementary topological properties of R, Q and Q
There is no loss of generality in restricting the proof to balls U(A; s)
where A is a rational number with the canonic series
A= 'Z Amgm
m= - f
as in the proof of Lemma 5. The general element x of U(A; s) has then
the canonic series
s — 1 00
*= Z 4.0" + Z ^0^
n = — f m = 0
here both the coefficients ^4m and xm are digits 0, 1,..., g — 1.
Let now {an} be an arbitrary sequence of elements of U(A; s). Its
elements an have by the last formula the form
s — 1 00
a„= Z Am9m+ Z amnQm + s (n= 1,2, 3,...),
m = — f m = 0
where the new coefficients am„ are again digits 0, 1,. .., g — 1. We call
amn the mth digit of an. It is again essential that each of these digits has
only finitely many possible values, 0, 1,..., g — 1.
On account of this property, it is possible to select successively in
{an} an infinite sequence of subsequences
*k = K°} (* = 0, 1, 2,...)
with the following properties,
(i) Z0 is a subsequence of {an}, and Ik is a subsequence of Ik _ x for
k>\.
(ii) For every suffix k > 0 the /cth digit of every term of Ik, and
therefore also the /cth digit of every term of Zt where I > /c, have the
same value, a(k) say.
We finally form the subsequence
of {aj. From its construction, I* has the #-adic limit
s — 1 00
z ^»0m+ z a<my+s.
m = — f m = 0
Since £/(^4; s) is a closed set, it contains this limit, proving that U(A; s)
is a compact set.
We shall soon discuss the importance of compactness for the theory
of functions on the balls of R and Q

Ordered rings 95
8 Ordered rings
The real field R is not only a field with the valuation \a\, but it has the
further important property of being ordered. Here a ring K is said to
be ordered if for its elements x an order relation > can be defined
which has the following two properties.
(a) Every element x of K satisfies one and only one of the relations
x = 0 or x > 0 or — x > 0.
(b) If x and y are two elements of K satisfying x > 0 and y > 0, then
also
x + y > 0 and xy > 0.
(c) If xeK is distinct from 0, then
x2>0.
This is obvious from (b) if x > 0. If, however, — x > 0, then x2
= (-x)2>0.
The real field R is ordered because the sign > can be given the usual
meaning of greater than and then satisfies these conditions.
A ring K cannot be ordered if it contains finitely many elements xl9
x2,. •., xn distinct from 0 and satisfying
x{ + x\ + . . . + x2 = 0.
This follows immediately from (b) and (c).
Therefore the complex field C is not ordered; for
l2 + ;2=0 where i =^/^1.
Neither is the p-adic field Qp ordered. For by the Main Lemma of
Chapter 6 § 4, Q2 contains an element x #= 0 such that
- 7 = 1 - 8 = x2,
and Qp for p > 3 contains an element y #= 0 such that
l-p = y2.
Hence in Q2 the equation
7 terms
x2+I2+ 12+ ... +1^ = 0,
and in Qp for p > 3 the equation
p - 1 terms
y2 + 'l2 + l2 + ... + l2= 0,
are satisfied, proving the assertion.

96 Elementary topological properties of R, Qp, and Qg
More generally, if g > 2 is any integer, then the g-adic ring Qg is not
ordered. It suffices to prove this in the case when g = p1p2- ■ ■ Pk is a
product of k > 2 distinct primes; hence the largest of these primes, pk
say, is at least 3. Denote by yk the /vadic number for which
1 ~ Pk = yl> and Put
A, = <0, 0,..., 0, yk\ A2 = Az = ... = APk = <0, 0,..., 0, 1 >.
These #-adic numbers are all distinct from 0, but it is obvious that
A\ + A\ + . .. + ^=0,
proving the assertion.
As will be seen, this negative property of Qp and Qg has important
consequences for the theory of p-adic and #-adic functions.
9 Functions on a ring with a pseudo-valuation
Let again K be a ring with the pseudo-valuation w(a), and let (K, d) be
the corresponding metric space with the distance d(a, b) = w(a — b).
A function
where £ is a subset of K, associates with every point x of E a unique
point y = f(x) of K.
Such a function is said to be bounded on E if there is a positive
constant c such that
w(f(x))<c for xeE.
If further x0 is any point of £, then f is said to be continuous at x0 if
there exists to every positive number S a second positive number
s = s(S, x0) depending only on S and x0 such that
w(/(x) — /(½)) < S if xeE and w(x — x0)<8.
In the notation of balls this is equivalent to
f(x)eU(f(x0\ S) if xeE and xeU(X0, a).
Finally, / is said to be continuous on E if it is continuous at every point
x0 of E. A function which is not continuous at a point or on a set is
called discontinuous.
Continuity on a set is a rather weak kind of continuity. By way of
example let (K, d) be (R, d) where R is the real field and d(a, b)

Functions on a ring with a pseudo-valuation 97
= \a — b\, and where we choose for E the set Q of all rational
numbers. The function f :Q^R defined by
/(x)=l/(x-v/2)
is continuous on Q because -Jl is irrational, and hence the
denominator x — y/2 does not vanish on Q. However, f is not bounded
on Q. For like every real irrational number,-Jl can be approximated
arbitrarily closely by rational numbers r, and then \f(r)\ is arbitrarily
large.
There is an important further kind of continuity. The function
f : E -» K is said to be uniformly continuous on E if to every positive
number S there exists a second positive number s = s(S) which
depends on <5, but not on x or x0, such that
w(f(x) — f(xo))<^ if x> x0eE and w(x — x0)<e. (1)
The distinction between continuity on E and uniform continuity on E
lies thus in the dependence or not of e on the variable point x0 on E.
For functions on compact sets this distinction becomes void, and
the following theorem holds.
Theorem 1 Let Ebea compact subset of K, and let f :E-^Kbe
continuous on E. Then f is uniformly continuous and bounded on
E.
Proof First assume that E is compact, but that f is not uniformly
continuous on E. There exists then a positive number S such that,
however small the positive number e is chosen, the property (1) is not
true. Apply this fact successively for e = 1/n where n runs over the
positive integers. It follows that there exist two sequences {xn} and
{x°} of points of E such that
w(f(xn)-f(x°n))>S and w(xn - x°n) < 1/n (rc = 1,2,3,...). (2)
Next, by the compactness of £, we can select an infinite subsequence
{x°ri, x°r^ x°3,...}, where 1 ^ rx < r2 < r3 < ...,
of {x°} such that the limit
w-lim x°rn, = x0 say,
n -» oo

98 Elementary topological properties of R, Qp, and Qg
exists and therefore lies in E. Since {xn — x°} is a null sequence, also
w-lim xrn = x0.
n-* oo
By the continuity of f at x0,
w-lim f(xrn) = w-lim/(x°n) = f(x0),
n -> oo n -> oo
from which it follows that
lim W(f(xJ - /■«,)) = w(/(x0) - f(x0)) = 0,
n-* oo
contrary to (2).
Secondly, assume that f is not bounded on E. There is then a
sequence {xn} of points of E such that
lim w(/(xn))=oo. (3)
n -» oo
But £ is compact. Hence an infinite subsequence {xrn} of {xn} can be
obtained such that
w-lim xrn = x0
n -* oo
exists, hence, by the continuity of/at x0,
w-\imf(xrn) = f(x0\
n -» oo
contrary to (3). This concludes the proof.
10 Continuous functions on a closed interval in R
Let again i^ be the real field with the valuation \a\ and (R, d) the metric
space with the distance d(a, b)=\a — b\. Let further i :a < x < b be a
closed and therefore compact interval on the real line.
In any first course on calculus, one mainly considers functions
f :i-^R continuous or differentiable on such an interval i. Of the
theorems then proved, the following ones are particularly basic.
(a) If f is continuous on i, if further f(a) ^ f(b\ and c is any number
between f(a) and f(b), then the equation
f(x) = c
has at least one solution x on i.
(b) If f is continuous on i, then f assumes on i both its infimum
(greatest lower bound) and its supremum (smallest upper bound).

Further remarks on real, g-adic, and p-adic functions 99
Next let x0 be an interior point of i so that a <x0<b. The derivative
or differential coefficient at x0 is defined as the real limit
r,( , I- f(x)-f(XQ)
f\x0)= km
x -»x0 X Xq
if this limit exists. In the next three properties the assumption is that f
is continuous on i and differentiable at all interior points of i.
(c) (Rolle's theorem) If f(a) = f(b\ then there exists at least one
point x0 satisfying
/'(x0) = 0, a<x0<b.
(d) (Mean value theorem of differential calculus) There exists a point
xx such that
f(b) -f(a) = f'ixJib -a), a<xx< b.
(e) If f is continuous on i, and if fix) = 0at all interior points x of i,
then f is a constant.
The proofs of these five theorems make essential use of the fact that
the real field R is ordered. On the other hand, as we saw in § 8, neither
the p-adic field Qp nor the g-adic ring Qg can be ordered. It will
therefore not come as a surprise that, as we shall see later, the
theorems (a) - (e) have no analogues for p-adic or g-adic functions.
There is, however, one important property of real continuous
functions which carries over at least to p-adic continuous functions
(but not always also to g-adic functions!) and which plays a big role in
our theory. This is Weierstrass's theorem on polynomial
approximations of continuous functions.
(0 Iff is continuous on i, and if S is any positive number, then there
exists a polynomial P(x) with real coefficients such that
\f(x) — P(x)\<S for xei.
11 Further remarks on real, g-adic, and />-adic functions
It is sometimes useful to study real continuous functions not on the
compact interval i:a<x < ft, but on some non-compact subset E
which is dense in i. Let, say, E = i n Q be the set of all rational points
on i. Then a function f :E^R continuous on E need be neither

100 Elementary topological properties of R, Qp, and Qg
bounded nor uniformly continuous, as is obvious from the earlier
example f(x) = (x -^/2)- *.
Exactly the same difficulty arises with p-adic and g-adic functions.
The set J = {0, 1, 2,...} now is dense in Ip and Ig9 but it is not closed
or compact. A function may therefore be continuous on J without
being bounded or uniformly continuous. However, if f : J -► Qg is
uniformly continuous on J, then it can always be continued into a
function F:Ig-^Qg such that F(x) = f(x) for xeJ and that F is
continuous and therefore (Theorem 1) uniformly continuous on I
Since every point on I is the g-adic limit of a fundamental sequence of
elements of J, F is by its continuity uniquely determined from its
values on J. The next chapter contains a fuller discussion of this
relation. All this holds of course also in the special case when g = p is a
prime.
12 Uniform limits and uniformly convergent series
The notion of uniformity met in § 9 in connection with the continuity
of a function is also of importance with regard to limits, or
equivalently, with regard to the convergence of an infinite series of
functions.
Let again K be a ring with a valuation w(a). Let £ be a subset of K,
and let {/„(*)} be an infinite sequence of functions fn:E-^K.
Assume that the limit
w-\im fn(x) = s(x), say, (4)
n -» oo
exists at all points x of E. This means that to every positive number S
there is a positive number N = N(S, x) depending on both S and x
such that
w(/n(x) - s(x)) <S if n > N.
If the number N is independent ofx for all x on £, then the limit (4) is
said to be uniform on E.
Not essentially different is the definition of a uniformly convergent
series. Let the elements fn(x) of the sequence {/„(*)} above have the
special form
n
fn(X) = Z ak(X)>
fc= 1

Uniform limits and uniformly convergent series 101
where the an:E-^K are given functions on E. The limit (4) now
becomes the infinite series
00
s(x) = w- Z an(x).
n= 1
If the limit is uniform on £, then the series is said to be uniformly
convergent on E. This will be the case if and only if to every positive
number S there exists a positive number N = N(S) independent ofxeE
such that
wl Yj ak(x) ~ s(x) I < <5 for n>N, xeE. (5)
Similarly as in real analysis, the following test holds.
Theorem 2 Let
00
n= 1
be a convergent series of positive real numbers cn, and let {an(x)} be
a sequence of functions an:E-^K such that
w(an(x)) < cn for xeE (n = 1, 2, 3,...).
00
Then w- Z an(x) converges uniformly on E.
n= 1
Proo/ Since the series C converges, there exists to every positive S a
positive N = N(S) such that
00
£ cfc < S if ft > AT.
k ~n + 1
Hence for n> N and for all x on £,
/ « \ / 00 \ 00
w s(*) - Z a*(*) = w Z flfcW ) < Z wK(*))
\ fc = 1 / \fc = n + 1 / fc = n + 1
00
fe-n + 1
The importance of uniformly convergent series lies in the following
consequence.
Theorem 3 Let {an(x)} be a sequence of uniformly continuous
oo
functions an:E-^K, and let the series w- Z an(x) be uniformly
n= 1

102 Elementary topological properties of R, Qp, and Q
convergent on E. Then its w-sum s(x)also is uniformly continuous
on E.
Proof. Let S and N be as in (5). Since the functions an are uniformly
continuous on £, we can to the given S find a second positive number e
independent of x and y such that
w(ak(x) — ak{y)) < S/(N + 1) if x, yeE, w(x — y) < e
(fc = 1, 2,..., n\ N < n < N + 1),
and therefore under the same conditions,
Since
s(x) - s(^) = - ( Z afeW - sW) + ( Z afcCv) - s(y)
\k=l J \k = 1
+ ( X ^ w - Z afcCv) i
\fe = 1 k = 1 /
it follows that
w(s(x) — s(};)) < S + <5 + S — 3^ if x, yeE, w(x — y)<s.
Since here ^ may be arbitrarily small, this proves the assertion.
13 Problems
1 Let K be a ring, and let
,, f0 if a = 0,
Wo(a) = |l if a #a
be the trivial pseudo-valuation of K. In the metric space (K, d) with
the distance d(a, b) = w0(a — b) show that every set is both open and
closed, and that a set containing at least two distinct points is not
compact.
2 Let L be an arbitrary field, t an indeterminate, and K the field of all
formal series
00
A= Z^n
n=f
where the coefficients An are in L and f is some rational integer. Put
w(0) = 0,

Problems 103
and if A / 0 and, without loss of generality, Af / 0, put
w(A) = 2~f.
Verify that w is a valuation of K which on L reduces to the trivial
valuation. The distance d(A, B) = w(A — B) makes K into a metric
space (K, d). Prove that every ball of K is compact if L is a finite field,
but that this is false if L contains infinitely many distinct elements.
3 Let p be a prime and w(a) = max (\a\, \a\p). Show that w is a pseudo-
valuation of Q. Put d(a, b) = w(a — b\ and denote by (Q, d) the
corresponding metric space. Is the ball w(a) < 1 in (g, d) (i) closed,
(ii) open, and (iii) compact?
4 Let p be a prime, and denote by /n: J -+Qp and gn: J -^ Qp the
functions
/n(*)= , 1 , n aild 9n(x) = ( (^2=1,2,3,...),
x + 1 + pn \X J
where
y\=1 hn\_pn(pn-1)(/-2)...(p"-x+1) for ^>x
fi J ' \x / 1 x 2 x ... x x "~
Decide whether the limits
lim fn(x) and lim #n(x)
«-» oo n-* oo
exist for xe J and whether they are uniform on J.
5 Let p be a prime. For real r denote as usual by [r] the integral part
of r, i.e. that rational integer [r] for which [r] < r < [r] + 1. For xe/p
put
00
/.(*) = I P*"0** (« = 1. 2, 3,.. .)•
k= 0
Find the largest ball L/(0; s) in which
p
lim /„(x)
n-> 00
converges uniformly, and determine the limit.
6 Decide whether the following g-adic series converge uniformly on
h-
oo / „^,\n oo
(gx)"
n= 0 n' n = 0
E ^f and £ ( 1 - |x|« )x».

8
First properties of continuous g-adic
functions
1 Introductory remarks
From now on, we shall be concerned mainly with the study of
continuous or differentiable functions
defined on the set I : \x\p < 1 of p-adic integers. However, since the
beginnings of this theory work without essential changes equally well
for g-adic valued functions
on the set Ig: |x| < 1 of g-adic integers, let us for the present consider
this more general kind of function. Even more generally, we shall
consider functions
defined on some subset of Ig. Here of particular importance are
functions
where
./={0,1,2,...}
is the set of all non-negative rational integers; such functions may also
be considered as infinite sequences
{/(0),/(1),/(2),...}
of g-adic numbers. Many properties can be obtained for quite
arbitrary functions of this kind.
The functions f :Ig-^Qgmay be considered as the analogues of the
functions/ : i -> R in real analysis which are defined on closed intervals
i:a<x<b. These closed intervals are identical with the closed balls
V(\(a + b\ \(b — a)) and correspond to the balls U(a; s) in Q which,

Operations with continuous functions on E 105
as we saw, are both closed and open. The linear transformation
x->(x — a)g~s
maps every ball U(a;s) onto the special ball
U(0;0) = Ig:\x\g<l.
Functions f :U(a;s)-^Qg on general balls U(a;s) are thus not
essentially different from functions on I On the other hand, the
g-adic integers in I have particularly simple canonic series.
2 Operations with continuous functions on E
Let E be any subset of Ig, x0 any point of £, and f :E-^Qga #-adic
valued function on E.
According to Chapter 7 § 9, applied with K = Qg, w(a) = \a\g, this
function /is said to be continuous at x0 if to every positive number S
there exists a second positive number s = e{S, x0) independent of x
such that
\f(x)-f(x0)\g<S if xeE and \x-x0\g<e.
Here, without loss of generality, it evidently suffices to choose for
S and a numbers of the form
S = g1~s and s = g1~t.
The definition just given can therefore be expressed in the following
equivalent form.
/ is continuous at x0 if there exists to every positive integer s a
second positive integer t = t(s; x0) independent of x such that
\f(x)-f(x0)\g<g-s if xgE and \x-x0\g<g-\ (1)
It will be in this form that we shall always define continuity.
Equivalent to this definition is the following one.
/ is continuous at x0 if and only if
9
lim f(xn) = f(x0) for every sequence {xn} of points ofE satisfying
n -*■ oo
9
limxn = x0. (2)
n -*• oo
Proof First assume that / is continuous at x0, and denote by {xn} an
arbitrary sequence of points in E with the limit x0. Then, as soon as n

106 First properties of continuous g-adic functions
is sufficiently large,
\xn - x0\g < g~l and therefore, by (1), \f(xn) - f(x0)\g <g~s.
Here s may be arbitrarily large, showing that the sequence {f(xn)} has
the limit f(x0).
Secondly, let/be discontinuous at the point x0. There exist then a
positive integer s and an infinite sequence {xn} of points of £ such that
9
lim xn = x0,
n-+ oo
while at the same time
l/(x„) ~ f(xo)\g >9~s (n = 1, 2, 3,.. .)•
Hence the property (2) is not satisfied.
Consider now two functions f :E-^Qg and /* :£->Q which are
continuous at x0eE. Then, by (2),
9 9 9
\imf(xn) = f(x0) and lim/*(xj = /*(x0) if limxn = x0.
n-*oo n-*oo n-*oo
(3)
By the laws for w-limits of Chapter 2 § 6, it follows that also
9 9
lim (f(xn) o /*(xn)) = f(x0) o /*(x0) if lim xn = x0
n -*■ oo «-*• oo
where the sign ° may stand for either +, —, or x . This proves:
Theorem 1 (i) If the functions f :E-^Qg and f* :E-^Qg are
continuous at x0e£, then so are f + f*, f — f*, and ff*.
(ii) If the functions fand f* are continuous on £, then so aref + /*,
f-f*,andff*.
Here, by the definition of continuity on £, the second assertion is an
immediate consequence of the first one.
By way of example, the two functions f(x) = c where c is a #-adic
constant, and/(x) = x are continuous on every subset £, as is obvious
from (2). On applying this property finitely many times, it follows
more generally that every polynomial
P(x) = P0 + P1x + ... + Prxr
with coefficients in Qg is continuous on every set £.

Quotients of continuous functions 107
3 Quotients of continuous functions
In the special case when g = p is a prime and therefore Qp is a field with
the valuation \a\p, the limit law for quotients of Chapter 2 § 6 shows
that Theorem 1 has the following analogue.
(i) If the functions f:E-> Qpandf* : E -> Qpare continuous at the
point x0eE and if /*(x0) =£ 0, then f/f* is continuous at x0.
(ii) If the functions fand f* are continuous andf* is distinct from 0
on E, then f/f * is continuous on E.
(4)
We wish to generalise this result to functions in the general ring Qg.
It suffices to consider the case when g = p1 p2... pk is the product of k
distinct primes pl9p2, - - - ,Pt where k > 2. As was found in Chapter 5
§ 5, Qg is the direct sum
Qg = QPl@QP2®.--@QPk
of the pK-adic fields QPk (k = 1, 2,..., k\ and it contains divisors of 0.
The ring Qg contains the subset Q*k of all elements with the
decomposition
< 0,..., 0,4c, 0,...,0), (5)
where the Kth component AK lies in QPk and the other components are
equal to 0. By the rules for addition, subtraction, multiplication, and
division, of Chapter 5 § 3 and 6, Q*k forms a field isomorphic to Q
We shall identify Q*k with QPk, thus shall not distinguish between the
g-adic number (5) and the /?K-adic number AK.
Assume now again, firstly, that the two functions f :E-^Qg and
f* :E-^Qg are continuous at the point x0 of E. These functions have,
say, the decompositions
/= </i,/2, ■■.,/*> and /* = </f,/J,...,/f>. (6)
If
EK = \oKl9 0k2, ..., oKk)
is defined as in Chapter 5 § 4, thus has the Kth component 1 and all
other components 0, then by (6),
and £(K)/ = <0,...,0,/K,0,...,0>
£<*>/* = <0,..., 0,/*, 0,..., 0>,
where the Kth components of/ and /* are in the Kth place and the

108 First properties of continuous g-adic functions
components are 0 elsewhere. Here E{K) is a constant, hence is
continuous at x0; therefore by the hypothesis and by Theorem 1 also
E(K)f and E{K)f* are continuous at x0. The same is then true of the
pK-adic components fK and /*.
We can now apply the property (4) and the rule of Chapter 5 § 6, for
the division of two g-adic numbers and arrive at the following result.
Theorem 2 (i) If the functions f :E-^Qg and f* :E-> Qg are
continuous at x0 e E, and iff* (x0) is not a divisor ofO, then f/f* is
continuous at x0.
(ii) If the functions fandf* are continuous on E, and iff* (x) is not
a divisor of0 for any x on E, then f/f* is continuous on E.
Here the assertion (ii) is again an immediate consequence of (i).
As an application consider an arbitrary rational function
R(x)=P° + PlX+-+PrXr, where Qsj=0
60 + ^ + --- + 6,^
and where all the coefficients Pp and Qa lie in Qg. By the remark to
Theorem 1 on polynomials and by Theorem 2, it follows immediately
that R(x) is continuous at all points x0 for which 20 + 2ixo + • • •
+ Qsxs0 is not a divisor of 0. If this denominator is not a divisor of 0 at
any point of E, then R(x) is continuous on E.
4 Some examples of discontinuous functions
In the last chapter we copied the terminology of real analysis and
called a function f discontinuous at x0 or on E if it was not continuous
at this point or on this set. Further the function is bounded or
unbounded on E according as to whether its g-adic value is bounded or
unbounded on E.
As a first example of a discontinuous function consider f :J->Qg
with
f{x) = l/(x - c),
where c = <cl9 c2,..., cfc> is a g-adic integer, g = p1p2- • • Pk being as
in the last section. If none of the pK-adic components cK of c lies in J,
then none of the components x — cK of x — c vanishes when x runs

Examples of discontinuous functions 109
over J, and therefore, by Theorem 2, f is continuous on J. However,/
is not bounded on J. For since c is a g-adic integer, we can find
elements x of J for which |x — c\g is arbitrarily small and therefore |/|
arbitrarily large. If one or more of the components of c lie in J, then
xeJ can be chosen such that x — c is a divisor of 0, and / is not
continuous at this point.
This function/can be continued to a function F : I -> Q0 by putting
fl/(x — c) if xe/g and x — c is not a divisor of 0,
[ 0 if xg/ and x — c is a divisor of 0.
By Theorem 2 this function is continuous at all points xelg for which
x — c is not a divisor of 0, but at points where x — c is a divisor of 0 it
evidently is discontinuous. It is clear that F is not bounded on I
The next examples are of a completely different kind and have no
analogues in real analysis. Denote by {an} = {a0, au a2,...} an
arbitrary sequence of #-adic numbers such that
9
ani=0 for all n, and lim an = 0.
n -*• oo
Thus {an} is a #-adic null sequence. We associate with this sequence
the two functions f :Ig->Qg and /2 :Ig-► Qg defined by
/l (*) =
and
/2(*)=
but xelg.
if x e J,
if x^J, but x£lg.
Both functions /x and /2 are discontinuous at the points of J because
for xeJ,
9 9
lim /x (x + gn) = lim ax + gn = 0, but /x (x) ^ 0,
«-*• oo «-*• 00
and similarly for /2.
Next let x be an arbitrary point of I which does not lie in J, and let
further {xn} be a sequence of distinct points of Ig with the limit x. As n
tends to infinity, f (xn) has the #-adic limit 0 independent of whether x
lies in J or not. On the other hand,/2 (xn) has in the first case the limit 0
and in the second case the limit 1. We obtain thus the result that

110 First properties of continuous g-adic functions
/i is discontinuous at all points of J and continuous at all points
of I that do not lie in J, and
/2 is discontinuous at all points of Ig, whether in J or not.
5 Uniformly continuous functions
Let again £ be a subset of I and let f : E -> Qg be a function on this
set. By Chapter 7 § 9, this function is called uniformly continuous on E
if there exists to every positive number S a second positive number
a = e(S) independent of x or x0 such that
\f(x)-f(x0)\g<S if x,x0eE and \x-x0\g<8.
It suffices again to take S = g1~s and s = g1~t where s and t are
positive integers, so giving the following equivalent definition.
f is a uniformly continuous on E if there exists to every positive
integer s a second positive integer t = t(s) independent ofx and x0
such that (7)
\f(x)-f(x0)\g<g~s if x,x0eE and \x-x0\g^g-'. (8)
By Theorem 1 of Chapter 7, such functions have the following
property.
Theorem 3 Every function f :I ->Q continuous on I is
uniformly continuous and bounded on Ig.
Proof By Chapter 7 § 7, the ball Ig = L/(0; 0) is compact relative to
the metric defined by d(a, b) = \a — b\g.
For many of the later considerations, the following theorem is
of importance, particularly in the special case when E = J and hence
E = Ig.
Theorem 4 Let E be a subset ofIg; let E be its closure; and let
f :E-^Qg be a function uniformly continuous on E. Then there
exists a unique function F :E-^Qg uniformly continuous on E and
bounded on this set such that
F(x) = f(x) if xeE.
Proof Denote by X an arbitrary point of E. There exists then a

Uniformly continuous functions 111
sequence of points {xn} of E such that
X = lim xn. (9)
(If X e £, then we may take xn = X for all n. Only the case when X¢Eis
of interest.)
Let now s be any positive integer, and let t = t(s) be a second
positive integer independent of x and x0 such that (8) is satisfied. By
(9), there exists a positive number N = N(t) such that
\xn-X\g<g-1 if n>N,
hence also
\xm-xn\g = \(xm-X)-(xn-X)\g^g-t if m,n>N,
and therefore, by (8),
|/(xm)-/(xn)|0<<Ts if m,n>N.
This means that {/(xn)} is a #-adic fundamental sequence. Let
L = lim /(*„) (10)
n -*• oo
be its limit.
77us /imft L does not depend on the special sequence {xn} used to fix X.
For if {x^} is a second sequence of elements of E such that
9
lim x'n = X,
n -*■ oo
then {xn — x^} is a null sequence, and hence there is a second positive
number N' = N'(t) such that
\xn-x'n\g<g-' if n>N\
hence by (8),
\f(xn)-f{x'n)\g<g-s ^ n>N'.
Therefore also {f(xn) — f(x'n)} is a null sequence, which shows that
also
lim /(*;) = L.
n -*• oo
Since then the limit (10) does not depend on the special sequence

112 First properties of continuous g-adic functions
{xn} tending to X, the function F :E-*Qg given by
g _ g
F(X)= lim f(xn\ whenever XeE and X= lim xn,
n -*■ oo n -*■ oo
where xneE for all n,
is well defined. It is obvious from this definition that F(x) = f(x) when
xeE.
Next, the function F is uniformly continuous on E. For let s and t be as
in (8), and let X and X0 be any two points of E satisfying
\X-X0\q<g
-1
Choose any two points x and x0 in E such that both
\x-X\g<g-1 and \x0-X0\g<g~\
and
\f{x)-F{X)\g<g-° and |/(x0)- F{X0)\g<g-\
It follows that
\x-x0\g=\(x-X) + (X- X0) - (x0 - X0)|, < g~',
whence, by (8),
\f(x)-f(x0)\g<g-s.
Therefore
\F(X) - F(X0)\g = | - (f(x) - F(X)) + (f(x) - f (x0))
+ (f(x0)-F(X0))\g<g-\
proving the uniform continuity of F on E.
Finally, F is bounded on E. For otherwise there exists an infinite
sequence {Xn} of points of E such that
lim|F(Z„)|9=TO. (11)
n -*• oo
Now E and hence also E are subsets of the compact set Ig. Hence there
is a subsequence
{Xri, Xr2, Xr3,...}, where 1 < r1 < r2 < r3 < ...,
of {Xn} such that the limit point
9
X0 = lim Xrn
n -*■ oo
exists. Since the points XYn lie in E and £ is a closed set, also X0 lies in
E. Now F(X)is uniformly continuous on E and therefore in particular

Operations with functions 113
is continuous at the point X0. But this implies that
\imF(Xrn) = F(X0),
n -*• oo
contrary to (11).
There cannot be a second function F* with the same properties as
F. For then F — F* would be uniformly continuous on E and
identically 0 on E. Since E is everywhere dense in F, the continuity
property would then imply that F — F* is also identically 0 on E. This
concludes the proof.
On applying Theorem 4 with E = J, we come to the following
consequence.
Let f :J -^ Qg be uniformly continuous on J. Then there exists a
unique function F :I ->Qg which is continuous and hence
uniformly continuous and bounded on I and has the property that
F(x) = f(x) if xeJ. (12)
This follows because J — I .
We call F the extension offto Ig and f the restriction ofF to J. When
f :J-^Qg is not uniformly continuous on J9. but only on a certain
subset F of J, then Theorem 4 can still be used to extend f to a
uniformly continuous function on E. Here F is now a proper subset of
I , because otherwise F would be uniformly continuous on the whole
of I and so in particular on J, against our assumption.
6 Operations with uniformly continuous functions
Let F be again a subset of I and lot f :E-^>Qg and/* :E-^Qg be two
functions that are uniformly continuous on F. There exist then to
every positive integer 5 two further positive integers t = t(s) and
t* = t*(s) independent of x and x0 such that
\f(x)-f(x0)\g<g-s if x.xeE and \x-x0\g<g-1 (13)
and
\f*(x)-f*{x0)\g<g-s if x,x0eE and \x-x0\g<g-'*.
(14)
Denote by T= T(s) the larger one oit and t*; also T is independent of

114 First properties of continuous g-adic functions
x and x0, and (13) and (14) can be combined to
\f(x)-f(x0)\g<g-s and |/*(x) - f*(x0)\g<g~s
if x, x0eE and \x — x0\g<g~T. (15)
Next, the hypothesis implies by Theorem 4 that f and /* are
bounded on E and therefore also on E which is a subset of E. Hence
there is a further positive integer u such that
\f(x)\g<g» and \f*{x)\g<g» if xeE. (16)
Now for x, x0eE and |x — xo|0 <g~T, by (15),
l(/(*)±/*(*))-(/(x0)±/*(x0))|,
= |(/(x) - /(x0)) ± (f*(x)-f*(x0))\g<g-\
and by (15) and (16),
|/(x)/*(x)-/(x0)/*(x0)|,
= l/(x)(/*(x)-/*(xo))+(/(x)-/(xo))/*(xo)l9<0-s + ''-
Hence the following result holds.
Theorem 5 Let f :E-^Qg and f* :E-^Qg be uniformly
continuous on £, a subset ofIg. Then also f + /*, f — /*, andff* are
uniformly continuous on E.
The two functions f(x) = c (c a #-adic constant) and f(x) = x
obviously are uniformly continuous on any subset E of Ig. Hence
every polynomial
P(x) = P0 + P1x + ... + Prxr
with coefficients in Qg is uniformly continuous on E.
7 The uniform continuity of quotients of uniformly continuous
functions
The existence of divisors of 0 in Qg presents again some difficulties
when dealing with quotients of functions. However, the following
result can be established.
Theorem 6 Let f:E-^Qg and f*:E-^>Qg be two functions
which are uniformly continuous on E, and let F :E-^Qg and F* :

Quotients
115
E-^Qgbe their continuations to the closure E ofE. IfF*(X) is not
a divisor ofO at any point ofE, thenf/f* is uniformly continuous
on E and F/F* is uniformly continuous on E.
Proof By Theorem 4 both F and F* are uniformly continuous on E,
and it is clear that
F(x)/F*(x) = f(x)/f*(x) if xeE.
It is then sufficient to prove that F(X)/F*(X) is uniformly continuous
on E. For the quotient F(X)/F*(X) exists and is unique since F*(X) is
not a divisor of 0.
Let, on the contrary, F(X)/F*(X) not be uniformly continuous on
E. There exist then two sequences {Xn} and {X'n} of points of £ and a
positive integer s such that both
\(F(Xn)/F*(X„)) - (F(X'J/F*(X'H))\g > g~s (17)
and
(18)
lim|Z„-X;|9 = 0.
n -*• oo
Since I is compact and E c I , we can next find an infinite subset of
{Xn}> say
{Xr ,Xr,Xr,...}, where 1 < rl < r2 < r3 < ...,
1 '2 '3
such that
lim Xrn = X09 say,
(19)
«-*• 00
exists; here X0 lies in £. By (18), {Xn — X^} is a #-adic null sequence;
hence also
9
lim x;„ = x0.
n -*■ oo
By (17),
|(F(XJ/F*(XJ) - (F(X;„)/F*(X'J)\g > g~\
which is the same as
F(XJF*(XJ-F(X'JF*(XJ
F*{XJF*(X'J
>9 \
that is,
\F(Xrn)F*(X'J-F(X'rn)F*(Xrn)\gx
1
F*(XJF*(X'J
>g s,
9

116 First properties of continuous g-adic functions
Here, by continuity and by (18) and (19),
lim (F(Xrn)F*(X'J - F(X'rJF*(Xr)) = 0.
00
It follows therefore that
lim
n -*■ oo
l
■F*(XJF*{X'rJ °°' (20)
Suppose now that F*(X0) is not a divisor of 0 in Qg. The function
F* is continuous at X0, and therefore, by Theorem 2, the same is true
for 1/F*. Hence the equation (20) leads to a contradiction.
From Theorem 6, we deduce immediately the following
consequence.
if
P0 + P1x + ... + Prxr
R{x) = — —
is a rational function with coefficients in Qg such that 60 + 61* + • • •
+ Qsxs is not a divisor of 0 at any point of E, then R is uniformly
continuous on F.
8 Locally constant functions, and step functions
Let again F be a subset of Ig and/ : E->Qg a function on F.
Definition 1 f is said to be locally constant on E if to every point
x0 of E there is a positive integer s = s(x0) such that
f(x) = f(x0) if xgE and \x-x0\^<g~s (21)
It is obvious from this definition that/is continuous on E.
The following theorem expresses an essential property of compact
subsets of I .
Theorem 1 Let E be a compact subset of Ig, and let further
f :E-^Qgbea function which is locally constant on E. Then the set
{f(x)\xeE}
of all values assumed by f on E has only finitely many distinct
elements.

Locally constant functions, step functions 111
Proof. Let the assertion be false. There now exists an infinite sequence
{xn} of points of E such that no two of the function values
f(xn) (n = 1, 2, 3,.. .)
are equal. By the compactness of E, an infinite subsequence {xrJ of
{xn} can be chosen such that the limit
9
lim xrn = x0, say,
n -*• oo
exists and lies in E. Denote by s the positive integer corresponding to
x0 for which the relation (21) holds. Then
\Xrn~ X0\g^Q
for all sufficiently large n, while all the function values f(xrJ are
distinct. Hence a contradiction arises.
Theorem 7 means that the compact set E can be written as the
union
E= (j U(ak;sk) (22)
of finitely many disjoint balls U(ak; sk) such that/is constant on each
of these balls.
Definition 2 Let E be a subset of Ig not necessarily compact. A
function f :E-^Qg is called a step function on E if there exists a
positive integer t independent of x and x0 such that
f(x) = f(x0) if x,x0eE and \x-x0\g<g~' (23)
The smallest integer t for which this property holds is called the
order off.
It is clear from this definition that a step function on E is uniformly
continuous and also locally constant on E.
Conversely, ifE is compact and f is locally constant on E, then f is a
step function on E.
For let t be the largest of the integers sk in the decomposition (22) of £
into a union of balls. By the proof of Lemma 4 in Chapter 7, each ball
U(ak; sk) can then be written as the union of finitely many disjoint
balls U(bji t\ and the same is therefore true for E itself. Since /is
constant on each of the small balls U(bj; t), / is a step function on E.

118 First properties of continuous g-adic functions
We can express the property (23) in a more explicit form. Let
x = x0 + x1g + x2g2 + ...(g)
be the canonic series of x so that the coefficients xn are digits
0,1,..., g — 1. On putting
x — x q -r x i g ~\~. . . —r x* i g ,
x(0 is one of the non-negative integers
# = 0,1,2,...,^-1,
and
\x-x«\<g-\ (24)
Put therefore
E(N) = En U(N; t) (N = 0, 1,.. ., g< - 1).
Since the different balls U(N;t) evidently are disjoint, the same is true
for the subsets E(N) of E. Thus E has the partition
N= 0
into finitely many subsets E(N) such that/is constant on each of them.
For general sets E one or more of the subsets E(N) may be empty,
but this is not so if E = J or E = Ig. In these two special cases,
N = 0,1,..., g* — 1 is an element o(E(N), and if x lies in E(N), so does
x + g\ Hence every step function on J or on I has the periodicity
property
f(x-\-gt) = f(x) for xgE.
9 The approximation of uniformly continuous functions by
step functions
In real analysis, functions continuous on a closed interval can be
approximated uniformly and arbitrarily closely by real step functions.
An analogous result holds for g-adic functions.
Theorem 8 Let E be either J or Ig. A function f :E-^Qg is
uniformly continuous on E if and only if there exists to every
positive integer s a second positive integer t = t(s) and a step

Problems 119
function S :E-^Qg at most of order t such that
\f(x)-S(x)\g<g-s for xeE. (25)
Proof (i) Assume that /and S satisfy (25). Ifx0 is a second point of E
such that
I* — xo\g — 9 •>
then simultaneously
S(x) = S(x0\ \f(x)-S(x)\g<g-\ \f(x0)-S(x0)\g<g-\
hence also
|/(x) - /(x0)|, = |(/(x) - S(x)) - (f(x0) - S(x0))\g < g~ \
which proves that / is uniformly continuous on E.
(ii) Assume next that/is uniformly continuous on £, and denote
again by s and t = t(s) two positive integers such that
\f(x)-f(x0)\g<g~s if x,x0eE and |x- x0\g<g~\ (26)
Let x(t) have the same meaning as in § 8, and define a function
S:E-^Qgby
S(x) = f(x(t)) if xeE.
Then S evidently is a step function at most of order t. By (24) and (26),
\f{x)-S(x)\g = \f(x)-f(x«%<g-\
so that (25) is satisfied.
Theorem 8 will later be applied when we study the approximation
of continuous functions by polynomials.
10 Problems
In the first three problems, x is an element of Ig and is assumed to be
written in canonic form
x = x0 + x1g + x2g2 + ... (g\
where the coefficients xn are digits 0, 1,..., g — 1.
1 Decide whether the following functions are uniformly continuous
on J, or are continuous on Ig.
f(x) = x0 + x1x2;f(x) = P(x0,x1,...,x10) where P is a
polynomial in its arguments with coefficients in lg\ and
f 1 if x0 = 0
[x/x0 if x0 f= 0.

120 First properties of continuous g-adic functions
2 Are any of the functions in Problem 1 locally constant or step
functions ?
3 Which of the following two functions are (i) continuous;
(ii) uniformly continuous; and (iii) locally constant on J?
00 00
n= 0 n= 0
4 Let p1 = 2 and p2 = 3, # = p1p2 = 6. Discuss the points of
discontinuity of the function
f(x) = (x — a)/(x — b) where a = < 1, 2 > and ft = < 1, 0>.
5 Let f :Ig^> Qg be defined by
,n f 0 if x = 0,
JW ll/|x|, if x^O.
Decide whether / is (i) continuous, and (ii) locally constant on Ig.
6 Let f : J -► Q0 be defined by
„ = i x + n
Decide whether this series (i) converges, and (ii) converges
uniformly on J, and whether f is (iii) continuous, and (iv) uniformly
continuous on J.

The interpolation series of a g-adic
function
From now on, the theory of functions of a p-adic or more generally of
a g-adic variable will be dominated by the interpolation series
'X
Z an
n = 0 x"
for such functions. This is somewhat analogous to the use of Fourier
series in the theory of periodic functions of a real variable, or to that of
power series in the theory of analytic functions of a complex variable.
1 Some preliminary formulae
Let again g > 2 be a fixed positive integer. The set I of all g-adic
integers contains as a subset the set
./ = {0,1,2,...}
of all non-negative rational integers. If x, j, k, n lie in J, then the
binomial coefficients
V\ x(x — 1).. .(x — n + 1) / — x\ fn\ fj + k\ / x
n)~ n\ ' V n )9\k)9 { j / \j + k,
are rational integers, hence have at most the g-adic value 1.
It is useful to mention the formulae
)=0 if 0 < x < n, and ( ) = ——:—- if x > n > 0,
nj \nj n\(x — n)\
and the following special case of the binomial theorem,
The following property of the binomial coefficient I ) will soon
w
play an important role in our theory.

122
The interpolation series of a g-adic function
If neJ and xg! , then
x
n
<1.
(2)
Proof. As a polynomial in x with coefficients in Q and hence also in Q
9
the binomial coefficient
x
n
is uniformly continuous on f
(Chapter 8, Theorem 5). Let {xr} be an arbitrary sequence of rational
integers in J such that
lim y = x.
r -*■ oo
Then also
lim p
r -*■ oo V n
X
n
Here, by what has already been stated,
x.
n
<1
(r = 1, 2, 3,...),
whence
72
= lim
r -*■ oo
X.
n
<1.
&
as asserted.
2 The interpolation series for functions/: J-*QM
Denote by f : J -> Q0 an arbitrary #-adic valued function on J; this is
thus essentially an infinite sequence
{/(0),/(1),/(2),...}
of g-adic numbers. For the next considerations this sequence need not
be restricted in any manner.
Definition The finite sums
where neJ, are called £/ze coefficients of f.

The interpolation series 123
We shall write an</> for an whenever we wish to stress the
dependence of these coefficients on the function/
In the language of difference calculus, the coefficients an of/are the
successive differences
an = A"/(0)
of/(x) at x = 0. The nth difference of/(x) at x is defined by the
equation
A"/to = I (- !)"" * (fcW + *)> (3)
and if the difference in the variable is not 1, but an arbitrary #-adic
number h, by
K /(*) = I (- !)"" * U W + **)• (4)
We associate with f the so-called interpolation series
/*(x)= Y an( ) where xeJ.
Since
) = 0 if n > x,
nj
this infinite series reduces to the finite sum
x ' x
n = 0 W
and so is defined on the whole set J. Its explicit value can be found as
follows.
By (2),
/•»-.t(,i<-iK;v<"-'>)(:'
On putting j = n — k and therefore n=j + k, this becomes
/*(*) = I/O) I (-l)fc/7 + KW
j = 0 fc = 0
j 7V + fc.

124 The interpolation series of a g-adic function
where
(j + k\( x \ = (/ + *:)! x! = x!
V k )\j + k) j\k\ ij + k)\(x-j-k)\ j\k\(x-j-k)\
= x\ (x-j)\ = /x\ (x -j\
(x-j)\j\k\{x-j-k)\ \j){ k J
Hence
Therefore by (1),
/*(x) = /(x) if xeJ,
and it has been established that
n = 0 V7
Either series on the right-hand side of this equation is called the
interpolation series off We may again sum to infinity because all terms
with n > x are equal to 0.
3 Relations between/and its coefficients an
The g-adic pseudo-valuation is non-archimedean. Therefore, on
taking on both sides of the equations (2) and (5) the #-adic values, it
follows from
< 1 for k,n,xeJ
9
that
max \an\g< max \f(n)\g
0 <n <x 0 <n< x
and conversely,
max |/(n)|g< max \an\g;
0 < n <x 0 <n< x
here 'max' is an abbreviation for maximum. On combining these two
00
= Z an
n = 0
X
n
if xeJ. (5)
(-If
< 1 and
9

Uniqueness of the interpolation series 125
formulae, we obtain the equation
max \an\g= max \f(n)\g if xeJ. (6)
0 < n < x 0 < n < x
In this formula, we are allowed to let x tend to infinity. If 'sup'
stands for the least upper bound, this leads to the important equality
sup |aj,= sup |/(n)|, (7)
neJ neJ
which will frequently be applied on the following pages.
4 The uniqueness of the interpolation series for/
The interpolation coefficients an of / : J -> Qg were uniquely
determined by the formula (2) and implied the representation (5) of this
function. The question arises whether there might perhaps be a
second development
n = 0 W
of/into an interpolation series with different coefficients a'n. If this is
so, put
b„ = a„ — a'„ for neJ*
and then
n n n
'X
Z bn[ )=0 identically in xeJ.
n = o \n
However, this identity allows us easily to prove that bn = 0 and
hence a'n = an for all suffixes n. For otherwise there would be at least
one suffix n such that bn =£ 0, and therefore there would also be a
suffix n with this property which is as small as possible. But then, on
choosing x = n, it would follow that
K(n) = K = o,
contrary to the choice of n.

126 The interpolation series of a g-adic function
The following result has thus been proved.
Every function f :J -^ Qg allows one and only one representation
f(x)= £ <M ) for xeJ
as an interpolation series, and here the coefficients an are defined by
the formula (2). (8)
5 The generating series of the sequences {/(/*)} and {an}
A simple formal identity which goes back to Euler allows us to express
the relation between/and its coefficients an as an identity connecting
the two formal power series
00 oo
£ f(n)z» and £ anZ"
n=0 n=0
which are called the generating series of the sequences {/(n)} and {an}.
For this purpose denote by z and Z two indeterminates over Qg
which are connected with one another by the three equivalent
relations
^ = 7^, Z = t^~, (l-z)(l+Z)=l.
1 +Z 1 —z
Then the following property holds.
The pair of formulae (2) and (5) which connects f with its
coefficients an is equivalent to the formal identity
00 00
£ f(n)z- = (1 + Z) £ a„Z*. (9)
n = 0 n = 0
Proof By this identity,
oo oo / y \n oo
X anZ" = (l + Z)-1 I /Wj-y = 2 /(n)Z"(l+Z)—S
n = 0 n = 0 \1_,_Z// n = 0
where by the binomial theorem
(l+z)—'= E ,, z< = Z(-if : z*
because
fc = o\ ^ / fc = o V/c
T'H'>fr'

Applications 127
It follows that
00 00 00 /k 4- n\
Zanz-= x Z/(«)(-i)M T z*+».
n = 0 n = 0 k = 0 \ K J
On comparing here the coefficients of any power ZN, where Ne J, on
both sides of this identity, we find that
N /N\
which is the formula (2). In the same manner, by replacing not z by Z,
but instead Z by z, we can also obtain the formula (5).
6 Applications of the identity (9)
The identity (9) allows many applications. A few simple examples will
suffice for the present.
Replace z in it by z' = — z so that
z'=-lfz> Z=-TTP (1 + ^0(1 + 2) = 1.
These new relations are thus symmetric in z' and Z. In this notation,
(9) takes the form
00 00
X (-lff(n)z'" = (l+Z) ^ anZ«,
n=0 n=0
or equivalently,
00 00
X a„Z» = (l+z') X (-iyf(n)z".
n = 0 n = 0
These two identities show that the relation between the two functions
(—l)xf(x) and ax, where xgJ,
is completely symmetrical and hence implies the following result.
If the function f :J-^>Qg has the coefficients an, then the function
f° \J-^Qg defined by
/°(X) = (-l)X for xgJ
has the coefficients
a°n = (-lff(n). (10)

128 The interpolation series of a g-adic function
As another example, consider the function f :J-^Qg defined by
f(x)=l/(x + l) for xeJ.
Since — 1 = < — 1, — 1,...,—1> and its pK-adic components — 1 do
not lie in J, it follows from Theorem 2 of Chapter 8, that f is
continuous on J. But f is not bounded on J because
1/(0--1)1, = 0- (11=1,2,3,...).
Hence neither is/uniformly continuous on J.
Both the function values/(n) and the coefficients an in the present
example are rational numbers. We are therefore allowed to apply the
following considerations which involve real values of z and Z rather
than g-adic numbers.
By the definition of/
oo I oo „n
Z /(«)*"=- z -
n=0 zn=1n
which is identical with the logarithmic function
-(l/z)log(l-z).
Hence by (9),
■■■
00 1/Z\_1/ Z \ 1
J>-2-=TTz(tTz) "■"('-TTzJ — z1'*'^
so that by a second application of the logarithmic series,
£ £(- i)nzn
z «„z»= z 4rr-
n=0 n=0 "n-1
This formula shows that/has the coefficients
an = (-lT/(n+l).
Naturally, these coefficients are not bounded in Q
In explicit form, by the definition of an,
j^(-l)«-*Q(/c+ I)"1 =(- 1)"(/I + I)" s
a formula which belongs to difference calculus.
It further follows from the property (10) that the similar function
/"(*) = (-1)7(1+ x)
has the coefficients
a°„ = l/(n + 1).

Series with the property (N) 129
7 Interpolation series with the property (N)
We are not so much interested in general functions f : J -> Qg as in
functions on J which are uniformly continuous on this set. For
then, by Chapter 8, (12) such functions can be extended to
functions F \Ig-^Qg which are continuous and hence also uniformly
continuous on J = Ig.
We know already that for all x in J
/<*)= 5>"(*)where fl-=ti(-^¢)^-^-
Here the interpolation series for/is meaningful because all its terms
with n > x are equal to 0.
But this is in general no longer true when x is any number in I not
in J. Now the series converges if and only if its terms tend to 0,
lim an(* )=0. (11)
«-*• 00
In order to arrive at a simple result, let us impose the stronger
condition that the interpolation series for f converges uniformly in
xelg, hence that also (11) is satisfied uniformly for all x in Ig.
There exists then to every positive integer s a second positive
integer n0 independent of x such that
x
n
< g s if n>n0.
9
We assert that then (11) can be replaced by the stronger limit
formula
liman = 0. (12)
«-*• 00
For let n be any suffix satisfying n > n0, and put x = n + gr where r
runs over the successive integers. Since ( ) is a polynomial in x and
W
so a continuous function of x, it follows that
9 9 AA fyi
lim x = n, hence lim ( ) = ( 1=1
r -*• oo r-» oo Vv \^
and therefore
\an\g<g~s.

130 The interpolation series of a g-adic function
Here s may be arbitrarily large, and so (12) follows immediately.
Interpolation series which satisfy the limit relation (12) are said to
have the property (N) (N for null sequence), and we say the same for/
Our result states then that
if the interpolation series for f converges uniformly on Ig, then both
the series and the function f have the property (N). (13)
It is now basic for our theory that the converse is also true, as follows.
Theorem 1 If f :J-^Qg has the property (A/), then its
interpolation series converges uniformly on Ig and on this set defines a
continuous and therefore uniformly continuous function F :Ig-^Qg
such that
F(x) = f(x) if xeJ.
Proof If ne J and xelg, then by (2),
IQI -
The property (N), i.e. the relation (12), implies therefore the equation
(11) uniformly in xelg. This means that the interpolation series
for f converges uniformly on I Now the terms of this series are
polynomials in x and hence are continuous on I It follows that the
sum of this series, F(x) say, is continuous on I (Chapter 7,
Theorem 3), hence is uniformly continuous on this compact set
(Chapter 7, Theorem 1). That F(x) reduces to/(x) when x is in J is
obvious.
Corollary Iff :J -+Qg has the property (N), then f is uniformly
continuous on J since J is a subset of In.
When/does not have the property (N), then by the last consideration
the limit (11) cannot be uniform on I I do not know whether this
limit can hold on Ig non-uniformly. In § 11 a sequence {an} will be
constructed which does not have the property (N), but for which the
limit (11) exists on a dense subset of I which is disjoint from J.

Series with the property (W) 131
8 A function continuous on I6 which does not have the
property (N)
Choose for g the composite number # = 6 = 2x3. The variable x in I
has a unique representation of the form
x = x0 + gy,
where x0 is one of the six digits 0, 1,..., 5, and y is again an element of
Ig. Define a function/ :Ig-^Qg by the equation
f(x) = (-1)^0,
so that evidently
f(x) = f(x0) if \x-x0\g<l/g.
Hence f is uniformly continuous on Ig and therefore also on J.
When x lies in J, the definition of/can evidently be replaced by
f(x) = (-If.
It follows that/has the coefficients
Since — 2 is not divisible by g = 6, it follows that
\an\g= 1 if neJ.
Hence f does not have the property (N).
9 Interpolation series with the property (W)
A classical theorem by Weierstrass (Chapter 7, Theorem (f)) in real
analysis states that every real-valued function f :i-^R continuous
and hence uniformly continuous on a closed interval i :a<x <b can
be uniformly approximated by polynomials. More explicitly, there
exists to every positive number e a polynomial P with real coefficients
such that
| f(x) — P(x) | < e for all x e i.
In analogy to this result, it seems appropriate to introduce the
following definition.
The functions f :J-^Qg and F :Ig-^Qg are said to have the
property (W) (W for Weierstrass) if there exists to every positive

132 The interpolation series of a g-adic function
integer s a polynomial P with coefficients in Qg such that
\f(x)-P(x)\g<g~s for all xeJ, (14)
and
\F(x)-P(x)\g<g~s for all xelg,
respectively.
An immediate consequence of this definition is as follows.
If the function f or F, has the property (W), thenf or F, is uniformly
continuous on J, or on Ig, respectively. (15)
Proof Consider, say, the function f and the relation (14). As a
polynomial, P is uniformly continuous on J. Hence there exists to the
arbitrarily given positive integer s a second positive integer t = t(s)
independent of x and x0 such that
\P(x)-P(x0)\g<g-s if x,x0eJ and \x-x0\g<g-1.
Therefore, by (14), applied at both points x and x0,
\f(x)-f(x0)\t
= |(/(x) - P(x)) + (P(x) - P(x0)) - (f(x0) - P(x0))\„ <g~°
whenever x, x0eJ and |x — x0\^<g~\ proving that/is uniformly
continuous on J. The uniform continuity of F on I can be shown in
exactly the same way.
The result of the next section will establish that the converse of (15)
is not in general true.
10 The properties (N) and (W) are equivalent
The following theorem gives a necessary and sufficient condition for a
function to have the property (W).
Theorem 2 I^et f :J-+Qg {or F :Ig-^>Qg) be uniformly
continuous on J (or on lg). Thenf (or F) can on J (or on Ig) be uniformly
approximated by polynomials if and only if its coefficients an satisfy
the equation
9
lim an = 0.
n -*• oo
In other words, each of the properties (N) and(W) implies the other one.

n = 0 xW
N 'X
(N) and (W) are equivalent 133
Proof. It will be sufficient to prove the assertion for the function F.
Assume, firstly, that this function has the property (N). Then its
interpolation series
oo , v
i = 0
converges on I uniformly to F, which means that its partial sums
N
n = 0 \n,
converge uniformly to F as N tends to infinity. Since these partial
sums are polynomials in x, F has then the property (W).
Secondly, assume that F does not have the property (N). There
exists then a positive number c and a sequence I of arbitrarily large
suffixes n such that
Let now P be an arbitrary polynomial with coefficients in Q say of
the degree r. This polynomial can be developed into a finite
interpolation series
n = 0 \n,
with certain coefficients bn in Qg. It follows that
x\ £ /x\ £ /x
/(x)-P(x)= Z (an-fen)[ + X a.[ = X
c.
n = 0 \W/ n = r+l V^2/ n = 0 ^ W
where
\an — bn if 0 < n < r,
n [ an if n>r + l.
Restrict now x to the subset J oiIg and apply the basic formula (7).
It follows then that
sup|/(x)-P(x)|0=sup|cn|0> sup \cn\g
xeJ nej n>r+ 1
= sup \an\g>c>0,
n > r + 1
because the sequence I contains suffixes n greater than r and for these
\an\g>c. This proves that F even on J cannot be uniformly
approximated by polynomials.

134 The interpolation series of a g-adic function
On putting
d(f) = d(F) = lim ( sup \an\g
r-» oo \ n > r + 1
we also see that always, for every polynomial P,
sup\f(x)-P(x)\g>d(f)
xe J
and
sup \F(x) - P(x)\g > sup | F(x) - P(x)\g > d(F).
xela xej
Let us in particular choose for/(x) = F(x) the function defined in
§ 8 which, as was shown, does not have the property (N) and hence
does not have the property (W). Since in this example \an\g = 1 for
all n, it follows that
sup |/(x) - P(x)\g > sup \f(x) - P(x)\g > 1,
xelg xe J
for every choice of the polynomial P. On the other hand,/is uniformly
continuous on both J and I
We see from this example that the property of uniform continuity
on J or on I is insufficient to ensure that the function can be
expanded into a uniformly convergent interpolation series, or
equivalently that it can be approximated uniformly by polynomials.
By contrast, it will be established in the next chapter that when g is
a prime /?, p-adic valued functions that are uniformly continuous on J
or on Ip lalways have the properties (N) and (W). In the general case
when g is a product of at least two distinct primes, this result will
enable us to give a full characterisation of those #-adic valued
functions uniformly continuous on J or on I which have the
properties (N) and (W).
11 On the convergence of a special interpolation series
This section deals with a special interpolation series
00 / v
n = 0 v n
which has coefficients satisfying
lim sup \an\ = oo.
|0
«-*• 00

A special interpolation series 135
but which nevertheless converges on a subset S o(Ig which is dense in
Ig and which has no point in common with the set J.
The construction is based on the well-known polynomial identity
;)-t(v)U)
This formula may be used when x and y are any #-adic numbers, and n
is any element of J.
Denote by j any number in J, and put
00
X
n = 0
= -j+ I 0(2n+1)2.
Then x lies in Ig, but not in J. For if it were equal to a number h in J, we
should have the equation
00
h+j= E 9(2" + 1)2-
n = 0
Here the left-hand side is an element of J, while the right-hand side is
an infinite canonic series and therefore cannot lie in J.
Denote by S the set of all values of x as j runs over J. It has already
been shown that S n J = 0. To prove that S is dense in Ig, it suffices to
note that j can evidently be so chosen in J that the canonic series of x,
2
X = Xq ~r X>\Q v ^2^ ~r . . . ?
where the xn are digits 0, 1,..., g — 1, begins with an arbitrary finite
sequence of given digits.
Denote now by N a positive integer, and put
N - 1 oo
y=-j+ £ g(2n + 1)\ hence x-y= £ 0<2» + 1>2.
n = 0 n = N
We assume that N is already so large that y is positive. Then
0<y<Ng(2N-V2<g4N2 (17)
because
for all positive integers N since g > 2.
To x and y as just defined we apply now the formula (16) with
n = g4N\

136 The interpolation series of a g-adic function
It follows immediately from (17) that
Hence it suffices to sum in (16) over the integers k = 1, 2,..., g4N2,
For these values of k the two binomial coefficients
V"1^ and ( y 1
k-\ ) \n-k/
are #-adic integers and have at most the #-adic value 1. Further
k )~ k \ k-\
where the largest value of |(x — y)/k\ is obtained when k = g4N2. Since
\x-y\g = g-ANl-4N~\
it follows that x and n as defined have the property that
,-42V- 1
9
We define the coefficients an now by
x
n
<g-4N~K (18)
^ = ^-" if n = g^2 (JV = 1, 2, 3,...),
0 otherwise.
n
These coefficients an are not bounded, but by (18) for xeS,
n-+ oo
Hence the series
9 fX
lim an ( 1=0.
'X
i = 0
converges for all x in S, as was to be proved.
n = 0 ^
12 Problems
1 Assume the function f : J -^ Qg has the coefficients an, and c is a
g-adic constant. Show that the function
c*f(x)

Problems 137
has the coefficients
2 If f : J -» Qg has the coefficients an, find the coefficients of the
function
/* w = f (*)c* _ *(! -c)*/^ -fc)-
3 If f :J->Qg has the coefficients an, find the coefficients of the
function xf(x\ and determine the function on J with the coefficients
ann.
4 The function f :J-^Qa satisfies
9
9 9
lim f(n) = lim an = 0.
n ~* oo n -*■ oo
Show that/is identically 0.
5 If/ :J-*Qg has the coefficients an, show that formally
oo „n ao „n ao „n
E f(n)-;= E -7 E an-r-
6 The function f :J -+Qg satisfies the recursive formula
n- 1
f(x + n)= E ckf(x + k) for xeJ,
fc = 0
where n is a fixed positive integer, and the coefficients ck are #-adic
constants. Show that the coefficients an off satisfy an analogous
recursive formula, but with possibly other coefficients than ck.
7 Denote by {cn} an arbitrary sequence of #-adic numbers and by
f :J -*Qg the function defined formally by
00 00 (X?\n °°
Prove that/has the coefficients an = (— l)nf(n).

10
Characterisation of functions with the
properties (N) and ( W)
1 The basic theorem for /?-adic functions
For the present let g = p be a prime. As before, let J be the set of all
non-negative rational integers, and let Ip be the set of p-adic integers.
We shall give not one, but four different proofs of the following basic
result.
Theorem 1 Assume the functions f \J-^Qp and F :Ip-^ Qp are
uniformly continuous on J, and on Ip, respectively, and have the
coefficients an. Then
(N) lim an = 0
n -*■ oo
and
(W) the functions f and F can be uniformly approximated by
polynomials.
By the last chapter, the two properties (N) and (W) are equivalent, and
hence if suffices to prove one of these properties. It further makes little
difference whether these properties are established for f or for F since
we may always choose
for f the restriction of F to J, and
for F the extension of f to In.
Next it is useful to remember that if F is continuous on Ip9 it is even
uniformly continuous on this set by the compactness o(lp.
The first proof of Theorem 1 was given by Dieudonne (1944). It is
then appropriate that we begin with an account of his proof that F has
the property (W).
2 Characteristic functions of sets
Let E be any subset of Ip. The characteristic function xE:Ip^> Qp of E

The polynomials jn (x) 139
is defined by
fl if xe£,
Xe(x) = x(x\E) = [0 if XElpMtx^
By way of example, if
E= U Ek
k= 1
is the union of n subsets Ek of E which are disjoint in pairs, then
n
X(x\E)= ^ x(x\EJ.
k = i
We shall apply repeatedly one particular case of this formula. Let t
be a positive integer, and let N run over the pf integers
0, 1, 2,..., pl — 1.
Denote by E(N) the set of all p-adic numbers x for which
\x-N\p<p~f.
Thus £(JV) is a subset of Ip and is in fact identical with the ball U(N; t).
Any two of the sets E(N) are disjoint, and just as in Chapter 8 § 8,
N = 0
It follows then that
l = 'f X(*|£W) if *e/,. (1)
N = 0
This identity can be generalised as follows. Denote by S :Ip-^Qp
any step function of order t on Ip so that
S(x) = S(N) if xeE(N).
It follows from this easily that
5(x) = Pf S(JV)*(x|E(A0) if xg/p. (2)
N = 0
3 The polynomials /, (jc)
Dieudonne's proof will be based on a number of simple lemmas.
Lemma 1 Let n and r be positiveintegers, and let xl9 x2,..., xr

140
Characterisation of functions
be r p-adic integers satisfying
\xp-l\p<p-"
Then also
(p = 1, 2,..., r).
EK -1
=i
<p
— n
Proof The assertion is true for r = 1; assume then that r > 2 and that
the assertion has already been proved for r — 1 factors. Hence
-1
ru -i
= i
<p n and |xr—1| <p n.
Therefore
r- 1
f| xp = 1 + pny and xr = 1 + pnz,
= l
where y and z are certain p-adic integers. Since then also y + z + pnyz
is a p-adic integer and since further
r
n *P = (i + pn>0(i + z>n*) = i + pn(y + z + z?n>>4
p = l
it follows that the assertion is true also for r factors and hence is
always true.
From number theory we take the following special case of a
classical theorem by Euler.
Lemma 2 Let x0 be one of the digits 1, 2,..., p — 1, let n be a
positive integer, and let
Hpn) = (p-^)pn-1
be Eulefs function. Then
x^pn) — 1 is divisible by pn.
For x0 is prime to pn.
For convenience put from now on
</> = 0(p«) = (p-l)p«-i.
Lemma 3 For all primes p and all positive integers n,
(j) > n.

The polynomials jn(x)
141
Proof Since p > 2, by the binomial theorem,
</> > (2 - l)2n" x = (1 + l)n" 1 > 1 +
Next denote by jn(x) the polynomial
/,(x)=l-x*.
1
= n.
Lemma 4 If x is a p-adic number such that \x\ > 1, then
\j»(x)\p = Mi-
Proof The assertion follows immediately from |x^| > |1| .
Lemma 5 If \x\ < 1, £/ze?i
lin(*) ~ 1L < P" n «wd therefore \jn(x)L = 1.
Proo/ By the hypothesis,
so that by Lemma 3,
l*L<l/p,
\j*(x) — 11- = I — **L < P"^ <P"n, whence also |jn(x)l = 1.
Lemma 6 // |x| = 1, tten
|j„(x)|p<p-».
Proo/ The p-adic integer x can be written in the form
x = x0 + py,
where x0 is one of the digits 1, 2,..., p — 1, and y is a p-adic integer.
Hence by the binomial theorem,
- Ux) = (x0 + pyf - 1 = (x* - 1) + £ (*\xt - \py)k.
Here, by Lemma 2,
Y</> _ 1 I < /7 ~ n
(3)
Further, for k = 1, 2,..., </>,
>'
*$_i(w)*
* U-i1 ° y
<
0/

142
Characterisation of functions
¢-1-
since l 7 ), x0, and y are p-adic integers. Next, by the
definition of </),
cj)pk
(P - 1)P
n- 1 +fc
<p
— n
(4)
For since always k<pk,k is at most divisible by pk 1, and therefore
<- for fc = 1, 2,..., 6.
v P
The assertion follows now from (3) and (4).
4 The polynomials hH{x; t)
Denote by n and t two positive integers. For t=l put
and for t>2,
Here
e0-> el-> • ' • ->et- 2
denote t — 1 positive integers which will depend on n and £ and which
must still be chosen. It is obvious that the products hn(x; t) are
polynomials in x with rational coefficients.
First let x satisfy the inequality
l*L<p '.
Then
max(\x\p,\p ^1^,...,1/7
t + 2x|P'IP"t + lxlp)<1'
and it follows from Lemma 5 that
\jn(p-kx)^l\p<p-n if fe = 0,l t_2,t-l.
Hence by the definition of hn(h; t) and by Lemma 1 applied with
r = eQ + e1 +... + et _ 2 + 1,
IM*;')-i|P^p~" ^ I4,<p-'. (5)
Secondly let 7 be one of the integers 0, 1,..., t — 2, £ — 1, and
assume that x satisfies the equation
\*\v = P '•

The polynomials hn(x; t) 143
Then
(<\ if fe = 0, 1 j-1,
\P~kAP <=1 if k=j,
(>1 if k=j + l,j + 2,...,t-l.
Hence, by Lemma 5,
\Jn(p-kx)\P = l if * = 0, 1 j-1;
by Lemma 6,
\jn(p-jx)\p<p-";
and by Lemma 4,
Un(p-kx)\p = Pik-J)* if k=j + l- 7+2,..., t-2, t-1.
On substituting these values for the factors of hn(x; t\ it follows that
\K(x;%<p~Uj\
where u- denotes the expression
Uj = nej - </>(e, + l + 2e. + 2 +... + (t -j - 2)et _ 2 + 1).
In the special case when j = t — 1, this formula must be replaced by
ut_1=n.
We assert now that we can choose the integers e0, eu ... ,^_2such
that also
u0 > n, uy > n,..., ut _ 2 > n. (6)
For this purpose denote by 0 the smallest positive integer which is not
less than (p — l)pn ~ 2, hence is not less than (j)/n = (p— l)pn ~ 2p/n.
(We are interested only in sufficiently large integers n) Then it follows
from the definition of u- that
Uj > n(ej -${ej + l+2£j + 2 + -.-+(t-j- 2)et _ 2 + 1}).
Let us now define the integers et _ 2, et _ 3,..., el9 e0 successively by
the equations
et_2 = (P + l,
et.3 = 0(et_2 + l) + l9
et_4 = $(et_3+~2et_2 + l) + l,
e, =0(e2+2e3 + ...+(t-3)et_2 + l) + l,
e0 = 0(e1 +2e2 + ...+(t- 2)et_2 + 1) + 1.

144 Characterisation of functions
It is then clear that the inequalities (6) hold, and hence we find that
\hn(x;t)\p<p-n if \x\p = p~{ where ./ = 0, 1,..., f-2, f-1..
(?)
Let again, in the notation of § 2, %(x\E(N)) be the characteristic
function of the set
E(N): \x-N\p<p-<.
The inequalities (5) and (7) can then be combined in the one inequality
\hn(x;t)-x(x\E(0))\p<p-n if xelp.
Hence, by a simple translation, it follows that for N = 0,1,..., pl — 1
also
\hn(x-N;t)-x(x\E(N))\p<p-n if xelp. (8)
Here n can be arbitrarily large. Our result implies therefore the
Lemma 7 All the characteristic functions
x(x\E(N)l where N = 0, 1, 2,..., pl ~ \
have the property (W).
It is now easy to complete Dieudonne's proof of Theorem 1. Let
F :Ip-^Qp be continuous and so also uniformly continuous on I
This implies that F is bounded on I say with the upper bound
M = .sup|F(x)|p.
xelp
Denote by s so large a positive integer that already
p~s <M.
By (14) of Chapter 8, there exists a second positive integer t = t(s) and
a step function S '.Ip-^Qp of order t such that
\F(x)-S(x)\p<p-s if xelp.
This inequality implies that also
sup|S(x)|p = M,
xe Ip
hence that
\S(N)\p<M for # = 0,1,...,^-1. (9)
Next, by (2), S(x) can in terms of characteristic functions be written

Algebraic numbers and integers 145
as
S(x)=P£ S(N)X(x\E(N)).
N = 0
Put similarly
H(x) = *X S(N)ha(x-N;t).
N = 0
Then #(x) is a polynomial, and
H(x) - S(x) = *% S(JV){As(x - AT; t) -|%(x|E(JV))}.
N = 0
It follows now from (8), applied with n = s, and from (9) that
\F(x)-H(x)\p<p-sM if xe/p.
Since in this inequality s may be chosen arbitrarily large, it proves that
F has the property (W).
5 Properties of algebraic numbers and integers
A second proof of Theorem 1 (Mahler 1958) is based on certain simple
properties of algebraic integers which can be found in any textbook
on the theory of algebraic numbers.
A real or complex number x is said to be algebraic (namely over the
field Q of rational numbers) if it satisfies some algebraic equation
goxi + g^-1+ ...+gd = 0, where 0O#O,
with rational integral coefficients g0, gl9... ,gd; the degree d may be
any positive integer. If x satisfies an equation of this form in which
Go = !,
then x is called an algebraic integer. Exactly the same definitions may
be given for p-adic algebraic numbers and integers, but will not be
required.
The following important statements are proved in the theory of
algebraic numbers.
Sums, differences, and products of algebraic integers are again
algebraic integers. (10)
If an algebraic integer lies in Q, then it is a rational integer. (11)

146 Characterisation of functions
6 Rational integral valued step functions that approximate
Let again f : J -> Qp be a function which is uniformly continuous on J
and hence also bounded on J. Since we may, if necessary, replace f by
puf where w is the integer defined by
p« = sup | f(x)\p9
xej
there is no loss of generality in assuming that
sup\f(x)\p<l.
xej
This means that not only the variable x, but also the function values
f(x), lie in Ip.
Let now again s be an arbitrarily large positive integer. There exists
then a second positive integer t = t(s) and a step function S : J -> Qp of
order t such that
\f(x)-S(x)\p<p-> if xeJ.
The different values S(N) of S(x) where N = 0,1,... ,pf — 1 are all /?-
adic integers. As we want to deal with rational integers rather than p-
adic integers, we select to each of these p-adic integers S(N) the
rational integer I(N) > 0 defined by
\S(N)-Z(N)\P<p-\ 0<Z(N)<ps-U
and then introduce a second step function Z : J -> Qp of order t by
I"(x) = 2;(JV) if \x-N\p<p~\ N = 0,1,. ..,^-1.
This new step function r has the convenient property that both its
variable x and its values I(x) are rational integers, hence no longer
involve p-adic numbers. It is clear that also
\f(x)-I(x)\p<p-s if xeJ. (12)
Denote again by an the coefficients oif, and similarly by bn those of
I; the coefficients of f — I are thenan — bn. The formula (12) implies
that
sup \an - bn\p = sup \f(x) - Z(x)\p < p~s. (13)
nej xe J
Our aim in the next sections will be to establish that
\bn\ <p~s ifnis sufficiently large. (14)

An explicit formula for bn 147
This implies by (13) that also
\an\p < p~s if n is sufficiently large,
and since here s may be chosen as large as we like, it proves that {an} is
a null sequence, hence shows that/has the property (N).
7 An explicit formula for bn using roots of unity
The step function I(x) is known completely when its pl values
Z(N), where N = 0,1,..., pl - 1,
are given; by our choice, these values are rational integers. It is now
possible to give a simple explicit expression for the coefficients bn in
terms of these values I(N) and of a p*th root of unity co, as follows.
Denote by
a primitive //th root of unity. Hence
CDpt=l, but q)^#1 if 0<t<*-1.
It follows that
pt ~ 1 {p
Z -m"= o
n = 0 lU
pl if pl\m,
if pl\m.
For this equation is obvious when pf\m, and it follows otherwise from
pt ~ l CDmpt - 1
1 ^ = -^=1=0-
n = 0 W l
Now put
K = Vl Z CD~mNI(N) (m = 0, +1,+2,...). (15)
N = 0
Then
'L^m' if 1?\m-m'.
Hence all coefficients Xm are known if the pf special coefficients
/i0, /l9 ..., Apt _ !
are given.

148 Characterisation of functions
The equations (15) can be solved for the values Z(N) in the form
m = 0 m = 0 N = 0
pt- 1 p* - 1
= p"' E ^(^ Z com(n-N) = i;(f2)
N = 0 m= 0
(0<rc<//-1).
Here both sides of this equation are periodic in n of period p\
It has thus been shown that I(n) allows for all n the representation
*(")= Z a
CO™
m = 0
On substituting this expression in
'n
K= Z (-1)^)^-/0,
it follows that
fen= Z (-in J I ^crf"*-*)
k = 0 \V m = 0
= Z ^ Z (-iW-K*-*),
m = 0 fc = 0 Vv
whence by the binomial theorem,
K=P*L ^,(0)--1)- (" = 0,1, 2,...). (16)
m= 0
8 The algebraic integer y
It will next be proved that the quotient
(co-\fp-l)pt~xlp = y, say,
is for t = 1, 2, 3,... an algebraic integer.
The proof depends on the well-known fact that the binomial
coefficients
'P\ P(p-i)...(p-k+i)
k)= lx2x...xfc ~ 0=1,2,....p-U
are divisible by p because their numerators, but not their
denominators, have a factor p.

Conclusion of the proof 149
Denote then by <fi (x) some polynomial in x with integral coefficients
which may be distinct from place to place. By the remark just made,
{x + \)p = xp + \+p(f){x\
(x + \)p2 = (xp + 1 + p4>(x)Y = (xp + \)p + p<j){x) = xp2 + 1 + P(j)(x\
and generally,
(x + Vf = x? + 1 + p<j>(x) ¢ = 0,1,2,...). (17)
For positive integral t the quotient
(x + \)pt - 1
Pt(x) =
^-1
(x+ir -l
is a polynomial in x with integral coefficients since
Pt(x) = (x + I^-Vp'-1 + (x + lfp~ 2)pt~x +... + (x + If'1 + 1.
By (17), this polynomial is the quotient
xpt + pcj)(x)
Pt(x) =
,t -1
xpt~l + p(j)(x)
and therefore has the form
Pt(x) = x(p-1)pt~1+p(l)(x) ¢=1,2,3,...). (18)
The //th root of unity co satisfies the algebraic equation
xpt-l
,t -1
= x^-1>'t"1 + x^-2>'t"1 + ... + x||t"1 + l=0.
x^"-l
Therefore co — 1 is a root of the equation
Pt(x) = 0
and hence is an algebraic integer. The same is therefore true for
0(0)-1). Since by (18).
Pt(o, -1) = (cd- \){p-l)pt~x+p<j){to - 1) = 0,
we obtain the assertion that
y=-</>(co-l)
is an algebraic integer.
9 Conclusion of the proof that | bn \p < p * for large n
By the representation (16),
p'K
(co-l)
pt ~1 (<am - 1V
-„= E p'm—r =^ sa^- ^19)
m = o \co — i /

150 Characterisation of functions
Here co and hence also the quotients
com - 1 _ f0 for m = 0,
0)-1 ~ jcom" x + of1 ~2 + ... + co + 1 for m = 1, 2, 3,...
are algebraic integers, and the same is true for the reciprocal
..-1 , vP* - i
CD = or
Next, by the construction, the function values
S(N) (# = 0,1,...,^-1)
are rational integers, hence also algebraic integers. By (15), the
products
pl x km (m = 0, 1,..., pf - 1)
are then also algebraic integers. It follows therefore from (19) that also
the expressions
Pn (72 = 0,1,2,...)
are algebraic integers.
Let now n be any suffix satisfying
n>n0, where n0 = (s + t)(p—l)pt~1.
Then
(co - 1)" = (co - l)n"(s + tHp ~ 1)pt ~' x (co - l)(s + mp ~ 1)pt"',
= oc(py)s + t say,
where the new number
a = (co-l)n-(s + 0(p-1)pt-1
is an algebraic integer because
n-(s+t)(p~ 1)^_1>0.
It follows now finally from (16) and (19) that
^~TT = P„ and therefore ^ = a/}„/ + '.
Since a, j?n, and y are algebraic integers, the second equation shows
that the same is true for the rational number bjps which is therefore a
rational integer. This implies that
\bn\p<P~s ^ n>n0,
but this is exactly the assertion (14) which was to be proved.

A proof using generating functions 151
10 A proof using generating functions
A simpler proof of the inequality (14) can be based on the identity (9)
between generating series which was obtained in Chapter 9. This
proof has the advantage that it runs entirely in the rational field, hence
that no properties of algebraic numbers are required.
Let again/ : J -> Qp be uniformly continuous on J, and let the step
function Z(x) and its coefficients bn be defined as in § 5. Our aim is
again the inequality (14). The new proof is based on the fact that I(x)
has the period p\
I(x) = I(x + pt) if xeJ.
As in Chapter 9, let z and Z be two indeterminates connected by the
three equivalent equations
Z = TTV> Z = Ti-' (1-z)(l+ Z)=1.
1 + Z 1 — z
Then by the general identity between generating series,
00 00
£ mzn = (1 + Z) I KZ-. (20)
n = 0 n = 0
Here, on the left-hand side, the suffix n may in a unique way be written
as
n = N + ptm, where 0<Af<//— 1 and me J.
It follows therefore from the periodicity of I(x) that
oo pt — 1 oo pl — 1
£ S(n)z" = £ S(N)zN £ z^ = £ 2(^(1 - z"*)" \
n = 0 N = 0 m=0 N = 0
so that by (20),
oo i p'-i / 7 \N / /7 V'N-1
.?.^-T^.?,Hrz)Hl!z)) •
or, say
^/- B(Z)'
where ^4 (Z) and £(Z) denote the polynomials
p*-i
A(Z)= ^ Z(N)ZN(l + Z)pt-N-\ B(Z) = (l + Zf-Zp\
N = 0

152 Characterisation of functions
which are both at most of degree pl — 1 in Z and have rational integral
coefficients.
Here, by the formula (17) of § 8 applied with x = Z,
B(Z) = (Zpt + 1 + p<j){Z)) -Zpt=l+ p<\>(Z\
where </>(Z) is a certain polynomial at most of degree p* — 1 in Z with
rational integral coefficients. Hence
00 A (7\ °°
£ ^z"=i Am= £ (-1)*-v-^z^zf-1. (21)
Here each product
is a polynomial in Z at most of degree Af(// — 1) with rational integral
coefficients. Therefore, on comparing the coefficients of the same
powers Zn on both sides of the identity (21), it follows that only those
terms
(_l^-i^-iy4(Z)0(Zf-1
can supply a contribution to the coefficient bn for which
N(p* - 1) > n,
and this contribution consists only of terms which are rational
integral multiples of pN~ *. We obtain therefore the result that
\bn\p<p-l{n-l)lipt-l» (22)
This formula shows again that the coefficients bn form a null sequence
and that therefore the inequality (14) is satisfied.
11 A proof based on recursive estimates
A fourth proof of Theorem 1 is due to R. Bojanic (1974) and depends
on recursive estimates for the interpolation coefficients. It is based on
an identity which goes back to difference calculus.
For our purpose it is convenient to derive this identity as a special
case of an addition formula for functions f :J-^Qp which for the
present can be arbitrary. Let as usual an be the coefficients of/so that
/w- e «.(;)= i«.(;) »«>■
Further denote by y din arbitrary number in J.

A proof based on recursive estimates 153
Also f(x + y\ where ye J, is a function of xeJ and as such has
certain interpolation coefficients, an(y) say, where
(.2 = 0,1,2,...), (23)
and then, conversely,
f(* + y)= t "*&)(*) = i<*n(y)(x) ^ xeJ. (24)
n = 0 \nj n = o W
Here the new coefficients an(y) have the explicit values
an(y)= lLan + k(k)= Z an+k[l) if y^J
{n = 0, 1, 2,...). (25)
Proof We apply the formula (16) of Chapter 9 which, with a slight
change of notation, we write as
X + y)=t(X)( " ) (- = 0,1,2,...).
m J „%\n)\m-n)
By the interpolation series for f(x),
m = 0 \ m J m = 0 n = 0\nJ\™-n;
=.?o(»)i.^(»-»)=.?•(») Joa"-(v'
from which, by the uniqueness of the interpolation series, the
assertion (25) follows immediately on comparing with (24).
Bojanic's identity is now
I «„ + *(£)= £ (-l)"~*(fcW + jO if y,»GJ, (26)
which is contained in (23) and (25).
Assume as before, that/ : J -> Qp is uniformly continuous on J, that
s is an arbitrary positive integer, and that £ = t(s) is another positive
integer such that
\f(*)-f(y)\P<P~s if *,yeJ and |x - )^ </?"'.
Then in particular,
|/(x+ //)-/Ml, <P~S for xeJ. (27)

154 Characterisation of functions
Since /is uniformly continuous on J, it is bounded on this set, and
hence it may without loss of generality be assumed that
\f(x)\p<l for xeJ.
By the general equation (7) of Chapter 9 it follows then that also
K\P<1 for neJ. (28)
Now, by (26), applied with y = p\
Here the previous result
(x^\)pt = xpt+\+p^{x)
means that all the binomial coefficients I J in the first sum are
divisible by p. Hence it follows from this equation and from (27) that
\an + ^\p<max(p-1\an+1\p9p-1\an + 2\p9...9p-1\an + ^_1\p9p-sy
(29)
We deduce immediately from (28) and (29) that
\an\p<p-1 for n>p\
On assuming next that n > pf and combining this estimate with (29), it
follows that
\an\p<p~2 for rc > 2//.
Repeating this recursive process s times, we finally obtain the estimate
\an\p<p~s for n>spl.
Since s may be chosen arbitrarily large, this once more proves the
assertion that {an} is a null sequence.
12 Decomposable #-adic functions
We return to the study of #-adic functions/ :J-^>Q and F :I -^Q
assumed uniformly continuous on J, or continuous and hence
uniformly continuous on I respectively. However, let now
g = plp2-.-Pk (k>2)
be a product of at least two distinct prime factors pl9 pl9..., pk. As we
saw in Chapter 9, these assumptions are insufficient to ensure that/or

Decomposable g-adic functions 155
F have the properties (N) and (W). Thus there remains the problem of
finding the additional necessary and sufficients conditions for (N) and
(W) to hold.
The new concept needed is obtained as follows. Let
Y— /Y(l) v(fe)\
be the decomposition of the #-adic variable x into its ^-adic
component x(1),. . ., its /vadic component x(k\ respectively, and let
similarly
/M = </(1)(x),...,/W(x)> and F(x) = <F<1>(xX...,F(*)(x)>
be those of/(x) and F(x). In general, each of these components/(K)(x)
and F(K)(x) will be a function of all /c components x(1),..., x(fe) of x.
Definition The functions/and F are said to be decomposable! if
for every suffix k = 1, 2,. . ., k,f(K)(x) and F(K)(x) depend only on
the Kth component x(K) of x.
With this terminology, our problem has now the following solution.
Theorem 2 Let f :J-^Qg be uniformly continuous on J, or
F :Ig-^Qg be continuous and hence uniformly continuous on Ig.
Then f or F, have the two properties (N) and (W) if and only if
they are decomposable.
Proof Since the proof is the same for/and F, it will be sufficient to
consider the function F.
First assume that F has the property (N), hence also the property
(W), and denote by
F(x)=„?/-(«) for xe/*
the interpolation series for F. Here {an} is a #-adic null sequence.
Therefore, if
an = {a^,...,a^y (30)
is the decomposition of an into its ^-adic component a{^\ . .., its pk-
adic component a£\ respectively, then the corresponding ^-adic
sequence {a(n1}},..., and the corresponding /?fc-adic sequence {a^} all

156 Characterisation of functions
are null sequences. Hence each of the interpolation series
F<-V>)= X a^[X (k=1, 2,..., k) (31)
n = o \ n J
converges uniformly for all x(K) in I and defines a (uniformly)
continuous function F(k) :In -+Q„.
The assertion is now proved if it can be shown that F(x) has the
decomposition
F(x) = <F(1)(x(1)),..., F(fe)(x(fe))> (32)
into a ^-adic component F(1)(x(1)),..., a /vadic component
F(k)(x(k)), respectively. For this purpose, let N be any positive integer.
The binomial coefficients ( ) are polynomials in x with rational
w
coefficients, hence have the decompositions
n
But then, for every positive integer N, also
N A,(l)\ n /uky
« = 0 \At/ n = 0 \"
Now, as AT grows indefinitely, the sum on the left-hand side tends
uniformly to the #-adic limit F(x), while the sums on the right-hand
side converge uniformly to the ^-adic limit F(1)(x(1)),..., the /?fc-adic
limit F(k)(x(k)). This proves that F(x) has the decomposition (32).
Assume, secondly, that F(x) is decomposable, say as in (32). Each of
the pK-adic components F(k)(x(k)) is (uniformly) continuous on the
corresponding set I and so, by Theorem 1, can be written as a
convergent interpolation series (31) where the coefficients ajlK) form a
pK-adic null sequence. This implies that the #-adic numbers an as
defined by the decomposition (30) form a #-adic null sequence.
Therefore
m= Z
'X
a.
n = 0 v n
is a continuous function on I with the two properties (N) and (W), as
was to be proved.
Theorem 2 severely restricts the class of continuous #-adic
functions that can be written as convergent interpolation series.

A non-decomposable function 157
13 An example of a non-decomposable function
The function f:I6 -► Q6 defined in Chapter 9 § 8 had coefficients an
satisfying the equation
|aj6 = l for xeJ
and so did not have the property (N). On the other hand, it is a step
function and therefore continuous on I By Theorem 2, it cannot be
decomposable.
This can be verified directly. Let
* = Oi,*2> and /(x) = </1(x),/2(x)>
be the decompositions of x and f(x) into their 2-adic and 3-adic
components, respectively. Here/was defined as follows. We wrote x in
the form
x = x0 + 6y,
where x0 is one of the digits 0,1,..., 5, and y is a number in /6. Then/
was given by
Write the components x1 and x2 of x similarly as
^1=^10 + 2^1 and ^2=^20 + ^2,
where x10 is one of the 2-adic digits 0 or 1, x20 is one of the 3-adic
digits 0, 1, or 2, yl is a 2-adic integer, and y2 is a 3-adic integer. It is
easily seen that x0 is given by the following little table in terms of x10
and x20-
X20
*io
0
1
0 1 2
0 4 2
3 1 5
This means that
(-l)xo =
+ 1 if x10 = 0,
— 1 if x10 = 1.
Since the value of f(x) = (— 1)*° lies in Q, this number has the
components
A(x) = /2(X) = (-If0.
Hence/2(x) depends only on the first component xl of x, proving the
assertion.

158 Characterisation of functions
14 Problems
1 Let f :J -^>Qpbe the function
f(x) = Fx for xeJ,
where Fn is the nth Fibonacci number defined by
F0 = 0, F, = 1, Fn + 2 = Fn+1 + Fn (72 = 0,1, 2,...).
Can the prime p be chosen such that f is uniformly continuous
on J?
2 The function/ : J-^Qg is defined by
/(x) = 5x for xeJ.
For which integers g>2 is f uniformly continuous on Jl Is f
decomposable for these values of gl
3 Let E be the subset of I6 consisting of all 6-adic numbers satisfying
|x-5|0<6
-2
and let %(x) be the characteristic function of £ in I6. Determine the
2-adic and the 3-adic components of the function
f(x) = x(x)(x-5) for xel6.
Is /uniformly continuous on /6? Is it decomposable?
4 E = {0, 2, 4,...} is the subset of all non-negative even integers in J,
and / : E -► Qp is defined by
* 'x
f(x)— Yj n\ f°r *e£.
n = 0 \W
At which points of E is /continuous?
5 The function/ :J-^>QP with the coefficients an satisfies
lim f(x) = 0.
n -*• oo
Show that the function
f°(x) = a2x for xeJ
is uniformly continuous on J.
6 The Bernoulli numbers Bn and the Bernoulli polynomials Bn(X) are
defined by their generating series
zez °° zn ze^ *+ ^z °° zn
7= T B„~- and — = V BJX)—,
C-l „^0 "n! e*-l Be0 " nV

Problems
respectively. It can be proved that for every prime p
\Bn\p<P (n = 0,
and that Bn(X) has the explicit form
Bn(X)= £^Bk(f\x»-k (n = 0,
Let x be an arbitrary element of I and {xn} a seq
elements of J such that
p
lim * =x.
n
n -*■ oo
Prove that
lim pXnBx(-
n -» oo \P/
exists and defines a continuous function of x on J

11
Further properties ofp-adic functions
It is convenient to collect a number of formulae connecting a function
f -J^QP with its coefficients an,
1 A first set of relations
We know that if the arbitrary function f :J -^ Qp has the coefficients
an=t (-ir*Q/(*) (.2 = 0,1,2,...), (1)
then, conversely,
/(*)= £ "Jfj (x = 0,1,2,...). (2)
It was also shown in the last chapter that, if ye J and
^)=1^^(^) (" = 0,1,2,...), (3)
then
"n(y)= io(-iy-k("j)f(k+y) (11=0,1,¾...) (4)
and
f(x + y)= t an(y) (*) (x, y = 0, 1, 2,..:). (5)
n = 0 W
To these formulae we may add, from Chapter 9 § 3, the pair of
equations
max \ak\p = max \f(k)\p in = 0,1, 2,...) (6)
0 < fc < n 0 < fc < n
and in the limit, as n tends to infinity,
sup|an(y)|p=sup|/(n)|1,. (7)
ne J nej
These formulae may be applied, with/(x) replaced by/(x + y\ and it

A second set of relations
161
follows then from (3), (4), and (5), that also for every integer y in J,
(8)
max \ak(y)\p= max |/(fc + 3>)|p
0 < fc < n 0 < fc < n
and
sup \an(y)\p = sup \f(n + y)\p = sup \f(n)\p. (9)
ne J nej n> y
Although we shall not use it, it may be mentioned that these
relations remain valid for #-adic functions on J; for the original proof
was given under this hypothesis.
We also recall from Chapter 9 § 6 that, if/has the coefficients an,
then the function
/°(x) = (-l)xax for xeJ
has the coefficients
a°n = (-l)nf(n) for neJ.
The effect of replacing/by f° in the equations (6) to (9) consists in
interchanging the two sides of these formulae.
2 A second set of relations
There is a second set of equations analogous to the equations (6) to (9)
which, however, holds only for p-adic functions because its proof
depends essentially on the property
\<*b\p=\a\p\b\p
of the p-adic valuation.
Evidently, the function/* :J^>QP defined by
/*(x) = f(x)- /(0), /*(0) = 0
has the coefficients
a$ = 0, a* = an for n > 1.
Hence
un
£ (- ir - * (I) aw - /(0)), /(*) - /(0) = t at (l
Here
n
k
n
k
x
/c-1
<
n
k

162
Further properties of p-adic functions
since l ) is a rational integer. It follows that
|a*|p< max
1 < k < n
n
^x(/(k)-/(0))
|/(n)-/(0)L< max
1 < k < n
n
x a
*
and therefore
i*
a
n
n
< max
p 1 < k < n
f(k) - /(0)
fin) - /(0)
n
< max
p 1 < k < n
a
*
On combining these two estimates, it follows that
f(k) - /(0)
max
1 < k < n
= max
1 < k < n
(10)
hence in the limit as n increases indefinitely,
f(n) - /(0)
sup
n> 1
n
n
(ii)
= sup
p n> 1
These two formulae may again be applied with f(x) replaced by
f{x + y) where ye J. This leads to the following pair of equations,
and
max
1 < k < n
sup
n> 1
k
an(y)
n
= max
p 1 < k < n
= sup
p n> 1
/('
f(k + y)~ fiy)
(12)
fin + y)- fiy)
n
(13)
If we subtract from/not the first terma0 = /(0) of its interpolation
series, but an arbitrary partial sum
N- 1
Z an
n = 0
X
n
of this series, then further sets of analogous formulae can be obtained.
In these formulae the denominator is not derived from ( J, but
from
x
N
Just as the relation (7) shows that the coefficients an are bounded if
and only if/(x) is bounded on J, so (11) makes evident that the

Continuity off at x = 0 163
quotients ajn are bounded for n > 1 if and only if the quotient
(f(x) — /(0))/x is bounded for x > 1.
3 Continuity of /at x = 0
As an application of the preceding formulae the question will now be
discussed how the continuity of an arbitrary function f : J -> Qp at
x = 0 is connected to properties of its coefficients an. To simplify the
discussion it will be assumed that
/(0) = 0 or equivalently a0 = 0.
We further use the notations
an= max \ak\p = max \f(k)\p
1 <k < n 1 <k <n
and
f(k)
which are valid by (6) and (11).
The following theorems will be proved in which n > 1.
Theorem 1 Assume that An\n\p^0as |n|p->0. Then f : J-+QP
is continuous at x = 0 if and only if
\an\p^0 as |n|p->0. (14)
Theorem 2 Assume that a„|n|p-> 0 as |n|p->0. Then f : J^QP is
continuous at x = 0 if and only if (14) is satisfied.
Theorem 3 Assume that f : J -> Qp is bounded on J. Then f is
continuous at x = 0 if and only if (14) holds.
Proof. We first note that Theorem 2 implies both Theorem 1 and
Theorem 3. For
ocn<An (n= 1,2,3,...),
and if / is bounded on J, then {a1? a2, a3,...} is a bounded real
sequence.
So let the hypothesis of Theorem 2 be satisfied, and let / be
^4n = max
1 < k < n
a,
= max
p 1 < k < n

164
Further properties of p-adic functions
continuous at x = 0. By (1),
|an|p < max
1 < k <, n
n
\f(k)
< max
p 1 < k < n
n
f(k)
(15)
Let s be an arbitrary positive integer. By the continuity of f at x — 0
there is a second positive integer t such that
\f(x)\p<p~s if |x|„<p-<.
We distinguish therefore in (15) between whether
\k\p<p~' or \k\p>p~t+1.
In the first case,
n
\f(k)
<P
— s
and in the second case,
n
f(k)
t-1
<p'-l\nf(k)\ Kp'-'-aJnl<p
— s
if
-s-t+1
This proves that (14) is satisfied.
Conversely, if (14) is true, then an inequality analogous to (15)
holds, withan replaced by f(n) and f(k) replaced by ak. The same kind
of proof shows therefore now that f is continuous at x = 0. For the
hypothesis (14) means that the function f°: J -> Qp defined by
/°(x) = (— 1)¾ is continuous at x = 0.
Theorem 2 is due to M. Waldschmidt (personal communication
1978), while Theorem 3 was given by H. Miiller (1977).
4 An example
The rational function
f(x) = x/(x + l)=l-(x+iy1
is continuous at all points of J, and in particular at x = 0. By Chapter
9 § 6 it has the coefficients
a0 = 0, and an = (- l)n+ 1/(n+1) for n>\.
For k = 1, 2, 3,...,
/V-1)=1-/>-*.

A further theorem by Waldschmidt 165
Hence f is not bounded on J, and Theorem 3 cannot be applied. But
neither can Theorem 1 or Theorem 2. For
apk-1= ±p~k
and therefore
Apk > apk > pk.
Hence, as n runs over the integral powers pk of p,
|n|p->0, but An\n\p>an\n\p>l.
This example makes it very clear that, even when f is continuous
not only at x = 0 but on the whole set J, no simple property of the
coefficients an can be deduced.
5 A further theorem by Waldschmidt
Waldschmidt (personal communication 1978) has obtained a further
result for functions continuous at x = 0 which shows that the p-adic
values of their coefficients cannot increase too rapidly.
Theorem 4 Assume that f :J -+Qp satisfies /(0) = a0 = 0. If f is
continuous at x = 0, then
|an|p/max(l,an)->0 as |n|p->0.
Conversely, if \an\p^0 as |n|p->0, then
|/(n)|p/max(l, «„)--*() as |n|p-»0.
Proof The second assertion is equivalent to the first one applied not
to f but to f°. It therefore suffices to prove the first assertion. We
proceed as in the proof of Theorem 2. Again as there,
<p~s if |/cL<p-<.
-t+ i
p
If, however, \k\ >p t+1 and therefore
n
k
<Pf >L
then
n
kjm
<Pf ^"UfifyL^P smax(l,an) if |nL<p'

166
Further properties of p-adic functions
Hence in either case.
n
f(k)
<p smax(l, an)
as soon as \n\ is sufficiently small, which is the first assertion.
6 A further example
The first assertion of Theorem 4 gives a necessary, but not also a
sufficient, condition for f to be continuous at x = 0, as the following
example shows.
Let f : J -> Qp be defined by its interpolation series
X
X
/(X)=?/«J'
n = 0
where the coefficients an are defined by
-1
a =
n
p l if n = p\ t = 0, 1, 2,...,
p-i^t-D ifn = ^-i,t=i,2,3,...,
0 otherwise.
This means in particular that again /(0) =a0 = 0.
By this definition,
and therefore
apt= max 1^ = p2' X>1 (¢=1,2,3,...)
i < * < pt
\an\ /max(l, an)->0 as |n| ->0.
On the other hand, the function f is not continuous at x = 0. For
pl / «* \ t / *j\ t
/W=I<J=IpX)+Ip
-(2j-1)i
n = 0
Here
i = o
A'-<?
j=i
y-i
= p'
because
-j(r
<pj<p' if 0 <j < t — 1.

The function j* 167
Further
It follows that
\f{V% = Vt for t= 1,2, 3,...,
hence that / is not even bounded in a neighbourhood of x = 0 and so
even less can be continuous at this point.
On considering instead of f the derived function /°, we find in the
same way that also the second assertion of Theorem 4 gives only a
necessary condition for \an\ to tend to 0 as \n\ tends to 0.
It may be noted that the function
/(x) = x/(x+l)
considered in § 4 satisfies the hypothesis of the first assertion of
Theorem 4. For since it has the coefficients
a0 = 0, an = {-\)n+1/(n+\) for n>l,
the number
an= max \an\p
1 < k < n
has the value
where pr denotes that integral power of p for which
(n+ l)/p<pr <n+ 1.
It follows that
aM > (n + 1)/P.
On the other hand,
|0„lp = 1 if l«lp<Vp>
whence
K\P/m3x(l> O < P/(" + l)->0 as |n|p->0,
as asserted.
7 The function/*
Let again f:J^>Qp satisfy /(0) = a0 = 0. So far, we have not
<
-(2t-l)i
P'-l
= \P

168 Further properties of p-adic functions
obtained a simple necessary and sufficient condition in terms of the
coefficients an for f to be continuous at x = 0. The following
construction leads to such a condition which, however, is not
particularly simple.
Since/ is assumed to be continuous at x = 0, there exists a positive
integer r such that
\f{x)\p<\ if \x\p<p~r.
Define a new function /* : J -> Qp by
/*(*) =/(Prx) for xeJ,
and denote by a* its coefficients. The quantity
a*= max \a*\p= max \f*(k)\p
1 <k <n 1 <,k <n
satisfies for all n > 1 the inequality
a* < 1.
Therefore both Theorem 2 and Theorem 3 lead to the result that
1**1,-0 as lnlj-0, (16)
and it can conversely be proved that this relation (16) implies that /*
and hence also f are continuous at x = 0.
The condition (16) can now be replaced by one for the coefficients
an. For
where
Hence a* has the explicit value
B /n\ ^ /nrk\ prk
<= Z(-ir'(J ZMm = Z^
fc = 1 W m = 1 \ m / m = 1
where cmn denotes the following sum,
The property (16) takes now the form
-+0 as 1^-+0. (17)
p
prk
I
m = 1
®mCmn

Continuity off on J 169
However, it does not seem to be easy to find a more explicit formula
for the coefficients cmn. That these are rational integers is evident.
8 Continuity off on J
If f : J -> Qp is bounded on J, there is no difficulty in formulating a
property of its coefficients an which holds if and only if f is continuous
at all points of J.
Firstly, it follows immediately from (5) and from Theorem 3 that f
is continuous at the single point y of J if and only if
KOOIp-^O as \n\p^0.
Hence f is continuous at all points y of j if and only if
\an(y)\p->® as Mp^O for every point y of J.
Now
an(y)= Z ak + n\l
On putting here successively y = 0, 1, 2,..., the following result is
obtained.
Theorem 5 If f :J -+Qpis bounded on J, then it is continuous at
every point of J if and only if
\ak + n\P^Q as M;,->0 for k = 0, 1,2,... (18)
For we find successively the limit formulae
Klp->0> K+an+l\p^>0> K + ^+1+^+2^0,...
as |n|p->0,
which implies (18) for the successive values of k.
The relations (18) need of course not hold uniformly in k. Here we
say that they are satisfied uniformly in k if there exists to every positive
integer s a positive integer t = t(s) such that
\ak + n\p<p~s if \n\p<p~\ /c = 0, 1,2,. .. .
Then every suffix m > 1 can be written in the form m = k + n where
k > 0, and n = pu is the largest integral power of p which is not greater
than m. Hence
|w|p->0 as m->oo,

170 Further properties of p-adic functions
and it follows that \am\ < p ~s as soon as m > p\ Since s may be chosen
arbitrarily large, this means that
|am|p->0 as ra->oo.
In other words, when the relations (18) hold uniformly in /c, then f has
the property (N) (see last chapter) and so is uniformly continuous on
J. The converse is naturally also true.
9 A third example
We next give an example of a function f :J -+Qp which is bounded on
J and continuous at all points of this set, but is not uniformly
continuous on J, thus does not have the property (N).
Write xe J in the canonic form
x = x0 + xxp + x2p2 + ...,
where the coefficients x0, x1, x2,... are digits 0,1,..., p — 1. Since x
is a non-negative rational integer, all digits xn with sufficiently large
suffix n are equal to 0, and so there exists a smallest non-negative
integer r = r(x) such that
xn = 0 if n>r.
This means that there also exists a smallest non-negative integer
m = m(x) such that
hence, if m > 1, that
xn = p— 1 for ?2 = 0, 1,..., m— 1.
In terms of the integer m so defined, put now
/(x) = (-ir(x).
The function / trivially is bounded on J. It further is continuous at
every point of J. For let x and y be two elements of J satisfying
\x-y\p<p-m-\
where m = m(x), and let
y = y0 + yiP + y2P2 + '-
be the canonic series for y. Then
yn = xn for w = 0, 1,..., m,

Two upper estimates 111
hence, by the definition of m and f,
m=f(x).
This proves the continuity of f at all points of J.
On the other hand, f is not uniformly continuous on J. For if it
were, then, by Theorem 4 of Chapter 8 there would exist a function
F :Ip->Qp continuous and hence uniformly continuous on I such
that
F(x) = f(x) if xeJ.
We show now that this is false by establishing that F is discontinuous
at the point — 1 of Ip.
Now the p-adic sequence
{p-l9p*-l9p*-l9...}
has the limit — 1. Its general element pk — 1 has the canonic
development
^-l=(p-l) + (p-l)p + ...+(p-l)^-i+0,
so that m(pk —l) = k and therefore
f(pk- 1) = (- If.
By the assumed continuity of F at x = — 1 we should have
lim/(/-l) = F(-l),
k -* oo
a formula which is false because the sequence {+ 1, — 1, + 1,
— 1,...} of the values f(pk — 1) has no p-adic limit.
A different example of a function which is bounded and continuous
on J, but not uniformly continuous, has been given by
M. Waldschmidt (personal communication 1978).
10 Two upper estimates for | f(x) —f(y)\p
It is obvious from the interpolation series
that if x and y are any two numbers in J satisfying x > y, then

172 Further properties of p-adic functions
For the binomial coefficients are rational integers and hence have at
most the p-adic value 1, and the first terms a01 I and a01 J cancel
each other.
This trivial estimate can be improved, as follows.
Theorem 6 Let f :J -+Qp be an arbitrary function on J'Jetx and
y be any two numbers in J such that x > y; and let N be a positive
integer. Then
l/W-/(y)lp
<max(m|ajp|x —y|p, \an\p\ l<m<N—l,N<n<x). (19)
(For N = 1 the terms with suffix ra, and for N > x the terms with suffix
n, are to be omitted.)
Proof Write
= fi(x, y) + f2(x, y), say-
Here for JV = 1 the first sum and for N > x the second sum are empty
Evidently,
l/W - f(y)\P < maxd/^x, y)\p, \f2(x, y)\p).
As above,
\f2(x,y)\p< max \an\p.
N <n< x
Next.
nj \ n j k = 0 y k i \n k
so that
/,<,.=;i\t(T)U
On the right-hand side evidently
max K/k\p<n\an\p.
1 < k < n

The upper bound w(t) 173
Since the binomial coefficients have again at most the p-adic value 1, it
follows then that
\fi(x,y)\p<\x-y\p max n\an\p.
1 <n<N- 1
The assertion is now obtained on combining these estimates for f1
and /2.
A very similar proof, which may be omitted, leads to the following
result which, instead of the coefficients an, involves the sums an(y)
from (3).
Theorem 7 Let f : J -*<2P be any function on J; let x and y be
two numbers in J such that x > y; and let N be a positive integer.
Then
\f(x)~f(y)\p<maxl\x-y[
flrntv)
m
Mn(y)V
l<m<N-l,N<n<x-yi (20)
(For N = 1 the terms with suffix m, and for N > x — y the terms with
suffix n, are to be omitted.)
In the special case when f has the property (N), thus when
p
lim a„ = 0,
n -> oo
either theorem shows that f is uniformly continuous on J.
11 The upper bound w (t)
Let again f : J -+ Qp be an arbitrary function on J, and put
w(t)=sup\f(x + pt)-f(x)\p.
xej
Here t may be any positive integer. The equation w(t) = 0 holds
exactly when
f(x + pt)=f(x) for xeJ.
Theorem 8 The value w(t) is finite if and only if f is bounded
on J.

174 Further properties of p-adic functions
Proof. It is obvious that w(t) is finite if f is bounded on J. Assume
next that w(t) is finite. If w(t) = 0, then f has the period // and so
certainly is bounded on J. We may then assume that
w(t) = p«,
where u is some rational integer.
Every number x in J can be written as
x = x(t) + jfy,
where x(t) is an integer satisfying
0<xa)<//-1,
and y is in J. Denote by v any rational integer such that
\f(x*)\p<pv for x* = 0,1,...,//-1.
Hence, if y = 0,
\f(x)\p<pv.
Exclude this trivial case so that y > 1, and put
xk = x(t) + /c// for k = 0, 1,..., y.
Then
y-l
_ v(0 -
X — X
2j (Xfe + 1 Xk)
k = 0
and therefore
\f(x) - /(x<% < max \f(xk+1) - /(xjl, < w(t) = p\
0 <k<y- 1
from which it follows that
\f(x)\p < max (|/(x«)|p, /7") < max (/7", /?*),
proving the assertion.
It is clear that if w(t) = 0, then also w(T) = 0 for T > t\ for if f has
the period p\ it has also the period pT. Furthermore, if f has the
period //, then it is uniformly continuous on J, and we have proved in
Chapter 10 § 10 that if
|/(x)|p^l if xeJ,
then its coefficients an satisfy the strong inequality
Let us exclude this case of periodicity, so that w(t) > 0 for all t. Now
the following statement holds.

The upper bound w(t)
175
Theorem 9 The function f is uniformly continuous on J if and
only if
limw(0 = O. (21)
f-*• CO
Proof If f is uniformly continuous on J, then to every positive
integer s there exists a second positive integer t = t(s) such that
\f(x)-f(y)\p<P~s if x,yeJ and \x-y\p<p-<.
But this implies that
w(0<P"s,
and since s may be chosen arbitrarily large, it proves (21).
Conversely, assume that (21) holds. There exists then to every
positive integer s a second positive integer t = t(s) such that
w(t) <p~s. Consider now any two numbers x and y in J for which
\x-y\p<p-\
The case when x — y is trivial; so assume that say x > y. Then x — y
has the form
x-y = np\
where n is a certain positive integer. Put
xk = y + ki? (k = 0, 1,. . ., n\
so that
*o = y> xn = x> and xk + l = xk + P* (fc = 0,1,..., n — 1).
Now
l/W-/001,=
V (/(¾+1)-/(¾))
fc = 0
< max \f(xk + 1)-f(xk)\p<w(t)<p s,
0 <fe <n- 1
which proves that f is uniformly continuous on J.
Corollary The limit relation (21) implies that the coefficients an
tend to 0 as n tends to infinity.
12 Estimates for the coefficients an in terms of w(t)
Let f : J->Q be any bounded function on J so that w(t) is finite for

176 Further properties of p-adic functions
every positive integer t. It was proved in Chapter 10 § 11 that
<w = -"i[ (¾+- + io (- i)"_ *(fc)(/(*+?') - /(fe))-
Here the binomial coefficients
(7 = 1,2,...,//-1)
are divisible by /?, and the p-adic values
\f{k + p')-f(k)\p
are at most w(t). It follows therefore that
k+^lp^max(p"1k+ilpJp"1k+2lpJ---Jp"1k+^-iLw(0).
(22)
We are assuming that / is bounded on J; more exactly, without
loss of generality let
|/(x)|p^l for xeJ
and hence also
|ajp< 1 for neJ.
Therefore, on applying (22) with t = 1, it follows that
\an + p\p<mzx(p-\w(l)) for n>0
and hence
\an\p < max (p~ 1, w(l)) for rc > p.
On combining this estimate again with (22), this time for t = 2, it next
follows that
\an + P2\P< max (p~ 2, p~ x w(l), w(2)) for n > p,
whence
\an\p<max(p~2, p~ 1w(l\ w(2)) for n>p + p2.
This procedure may be carried out k times, taking successively
t = 1, 2,..., /c, and it leads to the final result that
\an\p < max (/?-*, p1 -*w(l), /?2 _few(2),.. ., p~ lw{k - 1), w(k))
for n > p + p2 + .. (23)
Here k may be any positive integer.

A result by Ahlswede and Bojanic 111
13 A result by Ahlswede and Bojanic
We combine now Theorem 6 and the estimate (23) to prove a result
due to R. Ahlswede and R. Bojanic comparing the coefficients an with
the function w(t).
For this purpose it is convenient to make use of the Landau O-
notation well known from real analysis. Let X be a positive integral
variable, and let (j>(X) and \jj(X) > 0 be two real-valued functions of X
such that
\(j)(X)\<cil/(X) for all X,
where c is a positive number independent of X. Then one writes
4{X) = ow{X)).
In this notation, the following result holds.
Theorem 10 Let the function f :J -+Qp satisfy
\f(x)\p<l for xeJ.
Denote by a a constant satisfying
0<a< 1.
Then each of the two relations
w(t) = 0(p~M) for t>\ (24)
and
|fljp = 0(n-;) for neJ (25)
implies the other one.
Proof. First assume that (24) is satisfied. There is then a positive
constant c such that
w(t)<cp~at for £>1,
and here, without loss of generality,
c> 1.
It suffices to consider suffixes n satisfying n > p. To each such suffix
there is a unique positive integer k for which
pk + 2 -1
p + p2 + . .. + pk<n<p + p2 + ... + pk + 1= 1
/7-1

178 Further properties of p-adic functions
and therefore
pk + 2 > (p — \)n + p > n.
Now, by (23),
\an\p <max(p~k,p1~k-cp~a,p2~k-cp~ 2a,
...,p_1-cp"(fc"1)a,cp"te),
whence
\an\p<cp~ka
and therefore
\an\p<cp2n~\
which proves (25).
Conversely, let (25) be satisfied. We now apply Theorem 6, with x
and y replaced by x + pf and x, respectively. By the hypothesis (25)
there is a positive constant C such that
\an\p < Cn~a for n> 1.
The inequality (19) shows now that
w(t) < max (p" * - m\am\p, \an\p\ \<m<N — 1, N < n < x + pl).
Here choose
N = p\
Then, for 1 < m < N — 1,
p-t'm\am\p<p-t'N'CN-a = Cp-a\
and for AT < n < x + pf,
\an\p<CN~a = Cp-at.
These two estimates together prove the assertion (24).
Theorem 10 is only a special case of more general estimates by
Ahlswede & Bojanic (1975).
14 Problems
1 Let/ :J-^Qp be defined by
[ft if x = n2 is a square (neJ\
fix) = <
[0 otherwise.
Are there points x in J at which/is continuous?

Problems 179
2 Let f : J -+ Qp be defined by
,, vf 0 if x0 = 0,
HX) \x/x0 if x0^0,
where x0 is the first digit in the canonic series x = x0 + xxp
+ x2p2 +.. \p) of x, let / have the coefficients a„. Is it true that
\an\p<\n\p (« = 0, 1,2,...)?
3 £ is the set of all integers neJ which are not divisible by p, and
f : J -» Qp is defined by
m = 0 neE \" n,
Prove that/is continuous at x = 0.
4 Let / : J -> Qp be defined by the series
Show that/is continuous on J and that, for every positive integer t,
the series defines a uniformly continuous function on the subset
p~'<|x + l|p^l
of Ir
5 F :I -*Q is the function
00 /^X p
F(X) = Yj an\ b Wnere nm ^n = 0.
n = 0 \*V n-» oo
Determine the limit
p
lim JF(n!x)
n -»• oo
and prove that it is uniform on Ip.
6 Is the following assertion true?
If
supn\an\p<l,
neJ
then
00 /x
n = 0 V,
satisfies for all x and y in Jp the inequality
\F(x)-F(y)\p<\x-y\p.

12
Remarks on functions of two variables
We are interested mainly in functions of one variable as considered up
to now. However, in connection with the study of the differentiation
of such functions, it becomes necessary to extend some of the previous
theory to the case of functions of two variables.
1 The interpolation series for functions on J x J
Denote by J x J and Ip x Ip the set of all ordered pairs (x, y) of two
elements x, y of J, or of I respectively. We are concerned with
functions
f:JxJ-^Qp and F:IpxIp-^Qp,
where the variables are in either J x J or Ip x Ip, and the function
values lie in Qp.
Consider first the function/(x, y) on J x J. For fixed yeJ,f(x, y)
can be written as an interpolation series
/(*,30= tfm(y)h= ZfMl
m=0 \y/ m=0 \m
where the interpolation coefficients fm(y) are given by
These coefficients depend on the second variable y and have
themselves interpolation series
y /y\ £ (y
fm(y)= L amn\ = L Q
n = 0 \nJ n = 0 ^n
where
y 'n
amn= Z (-ir*(k)/*(*)
fc= 0
m n
= i £(-ir+"-*-*( jm/(M)- (i)
fc= 0 fc= 0

Uniqueness of the series 181
On substituting these series iorfm(y) in the development for/(x, y\ it
follows finally that
'**-.y.^Q-.y.*-(-Xi> ,2>
We call this development the interpolation series for j(x, y). Since this
series terminates after finitely many terms, there is of course no
problem of convergence.
2 Uniqueness of the interpolation series for f(x, y)
As in the case of functions in one variable (Chapter 9 § 4), the
question arises whether perhaps/(x, y) can not only be written as the
interpolation series (2) with the coefficients (1), but also allows a
second and different representation
valid for all pairs (x, y)eJ x J. On putting
Kn = amn ~ <*mn f°r K n)eJxJ,
such a hypothesis would imply that
£ ibmn(X)(y)=0 for all (XJ)€JXJ. (3)
m = 0 n = 0 \mJ W
We show now that this infinite system of homogeneous linear
equations for the coefficients bmn has the only solution
bmn = 0 for all (m,ri)eJxJ.
For assume that this assertion is false. There exists then a smallest
suffix M > 0 such that not all the coefficients
feMn, where n = 0, 1, 2,...,
are equal to zero. More exactly, there is then a smallest suffix N > 0
such that
But then
li>(XK(X)=^
contrary to (3).

182
Functions of two variables
This proof establishes that every function/ :J xJ -+Qp has one and
only one representation as an interpolation series, namely the
development (2) with the coefficients (1).
3 Relations between/and its coefficients
We proved in Chapter 9 § 3 the equations (6) and (7) which connect
the g-adic value of a function/of one variable with the g-adic values of
its coefficients. Exactly the same proof can be applied to functions in
two variables. It leads to the equation
max (\amn\p; 0<m<x,0<n<y)
= max (|/(m, n)\p; 0<m<x,0<n<y), (4)
and in the limit, as both x and y tend to infinity, to
suP(l^mnl;; ("*> ")e J x J) = sup(|/(m, n)\p; (m, n)e J x J). (5)
These formulae show in particular that if/is bounded on J x J, then
so are its coefficients amn, and vice versa.
There is a second pair of similar formulae, analogous to the
equations (10) and (11) of Chapter 11 § 2. The function/* : J x J -+Qp
defined by
/* (x, y) = f(x, y) - f{x, 0) - /(0, y) + /(0,0)
vanishes if at least one of the two variables x and y is equal to 0, and it
evidently has the interpolation series
X
f*(x,y)= £ £
a
Since
m = 1 n = 1
X
m
<
x
m
for m > 1,
mn
n
X
m
<
n
y
n
for n > 1,
we find, just as in the last chapter, that
a
max
"mn
mn
= max
; l<ra<x, 1 <n<y
f*(m,n)
mn
;l<ra<x, 1 <n<y
(6)
and in the limit, as both x and y tend to infinity,
sup
a
mn
mn
; m> 1, n> 1 1 = sup
/*(m, n)
mn
m > 1, n > 1 . (7)

J x J and Ip x Ip as metric spaces 183
4 The sets J x J and /^x / as metric spaces
In Chapter 7 § 2, we defined metric spaces, and in the remainder of
that chapter collected a number of properties of such spaces, dealing
mainly with metric spaces that were derived from rings with a pseudo-
valuation.
The more general spaces J x J and Ip x Ip are also metric spaces.
For if (x, y) and (x0, y0) are any two of their points, then the function
d((x, y\ (x0, y0)) = max (|x - x0|p, \y - y0\p) > 0
defines a distance between the points because obviously
d((x, y\ (x0, y0)) = 0 if and only if (x, y) = (x0, y0);
d((x, y\ (x0, y0)) = d((x0, y0\ (x, y));
and
d((x, y\ (x1? yt)) < d((x, y), (x0, y0)) + d((x0, y0), (x1? ^J).
All definitions and results of Chapter 7 can therefore be applied to
the spaces J x J and Ip x Ip. In particular, we can define continuity
and uniform continuity of functions, and uniform convergence of
series. We can further define functions /(x, y) that are bounded on
J x J or Ip x / or that are uniformly continuous in just one of the
two variables x and y.
It is again clear that J x J is everywhere dense in Ip x Ip.
5 Functions/: J x J-^>Qp which are uniformly continuous in x
So far,/could be an arbitrary function on J x J. It is of greater interest
to study functions which are uniformly continuous on this set. Let us,
however, first look at the more general functions which are bounded
on J x J and are uniformly continuous in just one of the two
variables, say in the variable x.
Since such a function can, if necessary, be multiplied by an integral
power of p, there is no loss of generality in assuming that/ :JxJ^Qp
has the following two properties.
|/(x,>0|p<l for (x,y)eJxJ; (8)
fis uniformly continuous in x uniformly in y; i.e. to every positive
integer s there exists a second positive integer t = t(s) independent of

184 Functions of two variables
x, x0, and y such that
\f(x,y)-f(x0,y)\p<p-s
if x,x0,yeJ and \x — x0\p<p~f. (9)
As we know, there exists to every xeJ a unique x0eJ such that
0 < x0 < pl - 1, |x - x0lp ^ P~ '•
With this choice of x0 = x0(x) define two functions /(1) :JxJ^Qp
and/(2):JxJ^gpby
/(1)(x,y) = /(x0,y) and /(2>(x, y) = f(x9 y)-f{1)(x, y).
Then, for all (x, y)e J x J, by (8) and (9),
f{1)(x + p>, jO = /(1)(x, y), |/(1>(x, y)|p < 1, (10)
and
l/<2)(x, y)\p = \f(x, y) - /(1)(x, y)\p = \f(x, y) - f(x0, y)\p < p~ \ (11)
Next, in analogy to the coefficients fm(y) in § 1, put
■m
/L1}(y)= Z (-ir"*( J/(1)(M),
fc = 0
m
/L2,(y)= I (-ir-h(^)/(2,(^y),
fc = 0
so that
L(y) = f^\y) + fZ\y). (12)
By (11) and by the formula (7) of Chapter 9,
|/^(>;)|p</7-s for m,yeJ.
Further, by (8) and (10), and by the formula (22) of Chapter 10,
\f^{y)\p< p-[{m- w"1)] for m,yeJ.
Hence there exists a positive integer m0 = m0(s) independent of y such
that also
\f^(y)\p<p-s if m,yeJ and m>m0.
On combining these two estimates for/J^y) and/J^y), it follows that
1/mMlp^P"5 if ™,yeJ and m>m0. (13)
Here/m(y) has the interpolation series
y ly
fm(y)= L amn
n = 0
n

Functions uniformly continuous on J x J 185
A second application of formula (22) of Chapter 10 leads therefore by
(13) to the result that
\amn\p<p~s for m>m0 and neJ. (14)
Since s may be chosen arbitrarily large, this is equivalent to the limit
equation
p
lim amn = 0 uniformly in ne J. (15)
m -* oo
6 Functions uniformly continuous on J x J
In analogy to the definition for functions of one variable, a function
f : J x J -> Qp is said to be uniformly continuous on J x J if to every
positive integer s there exists a second positive integer £ = t(s)
independent of x, y, x0, tod y0 such that
\f(x,y)- f(x0, y0)\p<p~s
if (x,y), (x0,y0)eJx J, and max(|x-x0|p, |y -y0|p)</7_t.
The following theorem gives a necessary and sufficient condition for
this property.
Theorem 1 The function f :JxJ->Qpis uniformly continuous
on J x J if and only if
|amn|p->0 as m+"n-» oo. (16)
Proof We first note that J x J is dense in /p x Ip. Next, /p x Ip is
compact; this can be proved by applying the method of Chapter 7 § 7
to each of the variables x and y. Thirdly, there is also a two-
dimensional analogue to Theorem 4 and its corollary (12) of
Chapter 8. Hence the uniform continuity of/on J x J implies that/is
bounded on this set. Since f may, if necessary, be multiplied by an
integral power of p, there is again no loss of generality in assuming
that (8) is satisfied. The function/further has the property (9), and it
also has the analogous property where x and y are interchanged.
Hence, by (15), \amn\p tends to 0 not only uniformly in n as m tends to
infinity, but also uniformly in m as n tends to infinity. From this (16)
follows at once.

186 Functions of two variables
Assume now that (16) holds. By formula (2) of Chapter 9,
<1 if (x,y)elpxlp.
m)\n
p
Therefore
x\(y
^mn
m \n
0 as m + n-> oo uniformly in (x, y)el x I
p
This shows that the series
FM=ioJoa-Q(n
converges uniformly on Ip x Ip. Its terms are polynomials in x and y
and so are uniformly continuous on Ip x Ip. Hence also F(x, y) is
uniformly continuous on Ip x /p. The same is then true for the
restriction/(x, y) of F(x, y) to the set J x J.
Let (x, y) be an arbitrary point of Ip x Ip. We can in many ways find
a sequence {(xn, yn)} of points of J x J such that
lim xn = x and lim yn = y.
n -* oo n -* oo
Then, by continuity,
p
F(x,y)= lim /(xn, yn).
n -» oo
Thus, in analogy to functions of one variable, the extension F(x, y) of
/(x, y) to Ip x /p is well defined (namely by the interpolation series)
when the values of /(x, y) at all points of J x J are given.
7 Problems
1 Discuss the convergence of the interpolation series
00 - x \(y
for (xj)eLx/
p p'
2 For (x, y)e/p x I write x2y3 as a canonic series
XV = Z0 + Ztp + Z2/72 + . . . (p).
Is the function F(x, y) = z5 continuous on Ipx Ipl

Problems
3 Let f : J x J -> Qp be defined by
/(x ) = {x if ix2+A^~3>
(0 otherwise.
Is /uniformly continuous on J x J?
4 Write the polynomial P :Ipx Ip->Qp defined by
P(x, y) = I 5
as an interpolation series.

13
The derivative of a function on J or I.
1 The derivatives f\y) and F(y)
The derivative or differential coefficient of a p-adic function is
analogous to that in real analysis, except that now the limit in its
definition is understood in the p-adic sense.
Consider arbitrary functions/ : J-> Qp and F : I -> Qp. Let x and y
for the first function be restricted to the set J, while for the second one
they may both be anywhere on Ip.
The derivative f' (y) of f at the point y e J and the derivative F' (y) of
F at the point yelp are now defined as the p-adic limits (x =/= 0)
/'00= Hm /(' + »-/0*
\x\p->0 X
and
p F(x + y)-F(y)
F'{y)= hm -^ ^ ^,
|xjp - 0 X
respectively, if these limits exist. Under this assumption it is clear that
the functions f and F are continuous at the point y.
Throughout this chapter, unless otherwise stated, it will be
assumed that/is uniformly continuous on J, and F is continuous and
therefore uniformly continuous on Ip. Moreover, f is to be the
restriction of F to J so that
f(x) = F(x) if xeJ.
Conversely, F is the extension of/to I Both functions have the same
coefficients
an= to(-W-k("j)f(k)= io(-l)n-k(f\F(k) (neJ)
and possess essentially the same interpolation series,
f(x)= JT an(X) for xeJ; F(x)= £ an[ ) for xel
V

The addition formula for F 189
Since both functions are uniformly continuous, by Theorem 1 of
Chapter 10,
(N) lim an = 0.
n -*■ oo
Hence the interpolation series for F is not only convergent, but is
uniformly convergent on I
In this chapter, we shall consider which properties of the
coefficients an can be deduced from the existence off'(y) or F'(y\ firstly
when y is a single point, and secondly when y runs over all points of J
and Ip, respectively.
2 The addition formula for F
We have already established, in Chapter 11 § 1, that, if x and y lie in J,
if/is an arbitrary function on J, and if an(y) denotes the sum
an(y)= Z fl*+»u} where neJ>
then
x ' x
n=0 V*2,
We prove now similar formulae for the function F continuous on I
in which, however, x and y may be arbitrary points on Ip.
By the property (N), the infinite series
an(y)= Z fl*+»u} where neJ> W
converge, and even converge uniformly on I and it is further clear
that
\an(y)\P->® as ft-»°o uniformly in y. (2)
Therefore the series
n = 0 \n,
converges uniformly in x and y on I . In explicit form,
j.*wG)-.y.*"b(0- ,3»

190 The derivative of a function
and here
ak + n\p^>Q as k + n^oo.
Now, by Chapter 10 § 11,
fc?0 \m - k) \k) { m )
It follows then by a permitted rearrangement of terms in (3) that
n=0 W m=0 \ m J
and so, finally,
'X
F(x + y) = X an(y)[ for x,yelp. (4)
n=0 \n/
In particular, when x = 0,
Fiy) = a0(y\
whence
00 /x\ °° x (x — 1
F(x + J»-F(y) = J>n(4„)= .1^(,,^
Therefore if x =j= 0,
F(x + jO-F(y)
x
and in particular,
t^f*-1) for
x,yelp, (5)
= 2l — 1 for x> yeJ- (6)
x n = i n \ n — 1
3 A lemma
The question arises whether in (5) and (6) the limit on the right-hand
side for |x| -> 0 may be obtained by substituting 0 for x in each term.
The proof will depend on the following lemma.
Lemma 1 Let f :J^>QP and F :Ip^Qp be as in § 1, and let
moreover
/(0) = F (0) = 0.

A Tauberian lemma 191
If
|/(x)/x|p->0 as |x|p->0, x^O, xeJ,
then also
|F(x)/x|p->0 as |x|p->0, x=£0, xelp.
Proof. The hypothesis is equivalent to the statement that to every
positive integer s i;here exists a second positive integer t = t(s)
independent of x such that
\f(x)/x\p<p~s if xeJ and 0 < \x\p<p~f.
Consider now an arbitrary point X of Ip satisfying
o<\x\p<P-\
If F(X) = 0, then
\F(X)/X\p = 0<p~s.
Excluding this trivial case, it may be assumed that
F(X)i=0.
The function F is continuous everywhere on Ip, hence in particular at
the point X. There exists then a positive integer u such that
\F(Y)-F(X)\p<\F(X)\p if Yelp and \Y-X\p<p~\
hence also
\F{Y)\p=\F{X)\p if Yelp and \Y-X\p<p~\ (7)
Since X is a p-adic integer, we can find to it a rational integer xeJ with
|x-X|p<min(/7-",|X|».
Apply now (7) with Y=x. Then
|x|p=|X|p and \F(x)\p=\f(x)\p=\F(X)\p.
Further, by 0 < \X\p < p~l also
0<|x|p </?"',
and so finally,
\F{X)/X\p=\f{x)/x\p<p-\
which proves the assertion because s may be chosen arbitrarily large.
4 A Tauberian lemma
The following result can now be proved.

192 The derivative of a function
Lemma 2 Let {au a2, a3,.. .} beap-adic null sequence with the
property that the limit
lim ta~(X~^ (xeJ)
o< \x\p -o„=i n \n-lj
exists and has the value 0. Then also {al/\, a2/2, a3/3,...} is a null
sequence, and the therefore convergent series
a.
Y (-l)n_1— is equal to 0.
Proof The function f : J -> Qp defined by
/(*)= Z an( J for xeJ
has by the hypothesis the property (N) and so is uniformly continuous
on J. Let
F(*)= £ a»(j for xgIp
be the extension of/to / ; also F is (uniformly) continuous on I . The
definition of/and the hypothesis imply that
/(0) = F(0) = 0.
Further, by the hypothesis and by (6), the expression
f{x)-/(0) = fjx) = ' ajx-i
x x n=1 n \n— 1
tends to zero for xe J, 0 < |x|p->0. Therefore, by Lemma 1, also
lim -^ = 0 if Xelp, X^0, (8)
0<|XL->0 %
Thus
/'(0) = F'(0) = 0.
The new function G :Ip^>Qp defined by
G(X)={ 0 for X = 0,
is continuous if X ^ 0, and it is by (8) also continuous at X = 0.

A Tauberian lemma 193
Therefore G and so also its restriction g :J^>Qp given by
g(x) = G(x) if xeJ
are uniformly continuous on Ip and J, respectively. They have then
both the property (N), and g(x) can be written as an interpolation
series
G(x)= Z M ) for xeJ,
n=0 W
where the new coefficients bn satisfy
p
lim bn = 0.
n -» oo
Next, evidently
F(X) = XG(X) and so also f(x) = xg(x).
Therefore identically,
x /x\ * . tx
Now
n=l \W/ n = 0 ^ W
*,»)=(x-B)u)+K»)=(B+1)(»+i)+,,u
so that
'^ Mr-kb\^%i)<
x ' X
= Z n(bn-l+K)
n= 1
Since the interpolation series of a p-adic function is unique, the
binomial coefficients ( ) must on both sides of this equation have the
W
same coefficients. This means that
an = n(bn _ ! + bn) (n= 1, 2, 3,...),
and therefore
lim — = lim (b„ _ ± + fcj = 0,
n -> oo W n -> oo
which proves the first assertion of the lemma. It also follows that for

194 The derivative of a function
every X in I the series
y ajX-X
n=i n\n-l
converges uniformly. Its limit as |X|p—>0 is then obtained by simply
putting X = 0 in all its terms, leading to
,xl-o„=i n \ n-\) ^^nXn-X nfi n
\x\P
Xelp
On restricting X to the set J, the assertion is obtained.
5 A necessary and sufficient condition for differentiability
The following basic result follows now easily.
Theorem 1 Let f :J^QP and F :Ip^>Qp be as in § 1. The
derivatives f'(y) and F'(y) exist if and only if
lim^O. (9)
n
n -» oo
If this condition is satisfied, then
00
f'(y)= Z (" ly1^ if yeJ
and
F'(y)= £(-1)--1^ if yelp.
Proof In the special case when f(y) = f'(y) = 0 or F(y) = F'(y) = 0,
respectively, the assertion is contained in Lemma 2 and its proof,
provided that the null sequence {an} in this lemma is replaced by the
null sequence {an(y)}.
In the general case, say that of the function F(x), assume first that
F\y) exists and consider the function
F*(x) = F(x) - (F(y) + (x - y)F'{y)).
Evidently
F*(y) = F(y)-F(y) = 0.

Condition for differentiability 195
Further
F*(x + y)-F*(y)
= (F(x + y)- F(y)) - {(F(y) + xF'(y)) - (F(y) + W(y))}
and therefore
F*(x + y)- F*(y) = F(x + y) - F(y) - xF\y\
so that
F*(x + y)- F*(y) F(x + y)- F(y)
x x
- F'iy).
As |x|p -> 0, the right-hand side tends by the assumed existence of F\y)
to the limit 0, and hence also
F*'(y) = 0.
Next, by § 2,
'X
F(x+y)-F(y)= £ an{y) x
and so
'X\ ., . /x
F*(x + y) - F*(y) = £ 0,(3,) I J - F'iy) ( x
The interpolation series for F*(x -f y) — F*(y\
00
F*(x + y)-F*(y)= Z a*(y)
n= 1
has therefore the coefficients
0*00 = 0100-^00, 0*OO = 0„OO for n>2.
Since F(x) is continuous on Ip9 the sequence {a„00} and therefore
also the sequence {a* 00} are nuU sequences. Again by the assumed
existence of F'(y\
0<|x|p-0 X M,-0,,= 1 n \n-l/
so that Lemma 2 can be applied. By this lemma, {a* (y)/n) and hence
also {an(y)/ri} are null sequences. Further
n= 1 W n=l W
which establishes also the second assertion. The case of the function/
is treated in the same way.

196 The derivative of a function
Assume, secondly, that the limit equation (9) is satisfied. Then the
series on the right-hand side of the formula (5) in § 2 converges
uniformly in x, and its limit for |x|p -► 0 is obtained by simply putting
x = 0 in all its terms. This limit being F'(y\ the derivative exists and
has the asserted development.
6 Differentiability at all points of J
Our first deduction from Theorem 1 is as follows.
Theorem 2 Let f :J^Qp and F :Ip^Qp be as in § 1. Then
bothff(y) and F'(y) exist at all points of the set J if and only if
lim^^ = 0 for all keJ. (10)
n -» oo n
If these conditions are satisfied, thenf'iy) has the interpolation
series
where the coefficients a'k are defined by
a
a
> £ (_iri_*±- for keJ. (11)
»=i n
Further
F'{y) = f'{y) if ye J.
Proof By Theorem 1, the existence off'(y) or F'(y) at all points of J is
equivalent to the conditions
p a (v)
lim -^ = 0 for all ye J,
n-* oo ^
that is,
0= lim 5l= lim a" + a"+1= lim a" + ^^+a-^
n -» oo n n -» oo n n -» oo n
n -*■ oo ^
and these equations, applied successively, evidently imply the
conditions (10) and are themselves implied by (10). It further follows that

Dieudonne's example 197
f'{y) = F'(y)= £ -—-— E «* + »
the series a'k converge. Moreover, by Theorem 1, for yeJ
n= 1 W fc= 0
The interpolation series for/'(j;) has thus the asserted coefficients (11).
In general, these new coefficients a'k do not form a null sequence
because/'(y) need not be uniformly continuous on J, and F'(y) does
not necessarily exist at any point of Ip that does not lie in J.
This is even the case when {a'k} is in fact a null sequence, as the later
example by Cassels will show. However, the following converse holds.
Theorem 3 Letf :J^QpandF :Ip^>Qpbeas in § 1. Assume
that the derivative F'(y) of F(y) exists and is continuous at all points
ofIp. Then the coefficients a'k defined in (11) form a null sequence,
and
Proof It is clear that F'(y) has a convergent interpolation series of
this form with the property (N), and it only remains to be proved that
its coefficients a'k have the form (11). This follows from Theorem 1 and
Theorem 2 because
n=l n n= X n k=0 VfC
-.?.(.?,<-v-,af)(0
As we are dealing with convergent series, the order of the terms may
be rearranged.
7 Dieudonne's example of a non-differentiable continuous
function on lp
In real analysis, Weierstrass gave the first example of a function that is
continuous on a closed interval, but without a derivative at any

198 The derivative of a function
interior point. As will now be shown, a similar existence result holds
for functions continuous on I ; such functions need not have a
derivative anywhere on I
We shall in the next section give an example by means of
interpolation series. However, the first example ever given is due to
J. Dieudonne (1944), as follows.
This example holds under the restriction that p > 3 so that the set
{0,1,2,...,/7-1}
of p-adic digits has at least three elements.
Consider any two elements x and y of I written as the canonic
series
x = x0 + xtf + x2p2 + . .. and y = y0 + yxp + y2p2 + ...,
where the coefficients xn and yn are p-adic digits. We define a function
F:Ip-+QponIpby
F(x) = xl + x\p + x\p2 + ...
and similarly,
F(y) = yl + y\p + y22p2 + .-.
This function is continuous on I For let t be any positive integer. If
\x-y\p<p~\
then
*o = 3^o» -^1 = 3^i9 • • • •> xt - l = yt- i'
hence also
\F(x)-F(y)\p<p-\
proving that F is uniformly continuous on Ip.
However, this function F has no derivative at any point x of Ip. In
order to prove this, it suffices to construct a sequence of digits
(3^ y^ yi> • • •} witn tne following properties.
(a) yn^Xn (72 = 0,1,2,...).
(b) If zn is the smallest non-negative digit such that xn + yn — zn is
divisible by /?, assume that znj^zn+1 for all suffixes n.
Since each yn has at least three possible values, these conditions can be
satisfied.

A non- differentiable function 199
Now put
/»> = x0 + x1p + ... + xll_1p"-1+3;llp" (n = 0,1, 2,.. .),
so that the sequence {y(n)} has the p-adic limit x. Evidently
and similarly,
F(x) - F(yW) = (x2n - y2n)pn + x2n+ lPn + 1 + x2n + 2p"+ 2 + .. .,
from which it follows that we obtain the canonic development
^r^.+^v*
,"„2
x-y'
where z', z",... denote certain digits. Here no two successive first
digits zn and zn + 1 are the same; hence the left-hand side has no p-adic
limit for n-> oo. This shows that F'(x) does not exist.
8 A non-differentiable function defined by its interpolation
series
Functions that are continuous, but not differentiable, on I can also
be constructed by means of Theorem 1.
The simplest example of such a function is given by
Hx)= lpr(X)= ian(X\
rf0 \P J n = 0 W
where the coefficients an have the values
(pr i(n = p\ re J,
n \o otherwise.
From this definition, {an} is a null sequence, and hence F is continuous
on Ip. For every point y in Ip,
so that in particular,
= LP
f r = s \pr - ps
= 1 + p(f + i_ps) + P\f + 2_f) + -'- = i + «s(y\

200 The derivative of a function
where ois(y) is a series which from its form satisfies the inequality
\as(y)\P<VP (5 = 0,1,2,...).
From this estimate it follows at once that {an(y)/n} is not a null
sequence, and hence, by Theorem 1, F\y) does not exist.
A small change in the function just defined leads to a function with
very different properties. Let us namely put
so that now
(pr if n = pr— 1, re J,
n [0 otherwise.
Again {an} is a null sequence, and therefore F(x) is continuous on I
By definition,
a„(y)= i>* + „(f)= I A r y ,
k = 0 VV pr>n+l \P n -1.
Consider first the case when y — — 1. Since for r, seJ,r> s,
r '! ^ = (- 1)^-^-1 =(- 1)^-^-1 = - 1,
it follows that
00 *>s + 1
r = s+ 1 P — A
Therefore
and so {an( — l)/n} is not a null sequence. Hence F'( — 1) does not exist.
Secondly let y ^ — 1. The identity
'y + l\ = l+l_/ y
y — n) pr — n\pr — n — l
shows that
1_ Pr(pr-n)/y + l\
+ VA+1 n \pr-n)'
n y +
Here the summation is extended over all re J for which pr > n + 1. It
follows that n is divisible by no power of p higher than pr ~ 1, so that

An example by Cassels
201
for all terms of the sum
p —n
n
<1.
Again the binomial coefficients have at most the p-adic value 1,
whence
*niy)
n
<
1
\y + il.
sup
pr > n + 1
P~r<
1
n\y + l
This estimate proves that {an(y)/ri} is a null sequence when y ^ — 1,
hence that F'(y) exists in this case.
In fact, if an arbitrarily small neighbourhood of the point — 1 is
excluded from I the convergence of this null sequence is uniform in
the remaining set, and hence F'(y) is continuous on this set.
Since, in particular, F'(y) exists for all ye J, all the series a'k converge
by Theorem 1. On the other hand, {a'k} is not a null sequence. For ak + n
vanishes except when k + n has the form pr — 1 for some re J. It
follows that
a
;= Z (-i)p-*-
fc-i
pr > k + 2 P
which means for k = ps — 1, where s is any positive integer, thai
\a„s^
11/7
= l/p.
This proves the assertion.
9 An example by Cassels
Theorem 3 states that if both F and its derivative F' are continuous on
Ip, then {a'k} is a null sequence and is the sequence of the coefficients of
F'.
This raises the question of whether the converse is true: if {a'k} is a
null sequence, does this imply that F' exists, is continuous on I and
has the coefficients a'k ?
Surprisingly, the answer is in the negative as was first proved by
J. W. S. Cassels (personal communication 1956) by means of an
example. Moreover, in this example, the non-zero coefficients an of F
itself do not tend too rapidly to 0.

202 The derivative of a function
Theorem 4 There exists a function F :Ip-^Qp continuous on I
with the following properties.
lim supn\an\p= 1. (12)
n -*• oo
All the p-adic series
00 a
<= £ (-l)""1-^ (£ = 0,1,2,...) (13)
n= 1 W
are convergent.
{a'k} is a null sequence. (14)
The derivative Ff does not exist at all points of Ip. (15)
Proof. The example of a function F with these properties is defined in
terms of its coefficients an, as follows.
Put
Uj = Pj +j and vj = pi+1 +j (j = 1, 2, 3,...)
so that all the integers Uj and v- are distinct and satisfy the inequalities
0 < u1 < vx < u2 < v2 < u3 < v3 < ...
If now the suffix n of an is not one of the numbers u- or vj9 let
an = 0.
Thus in particular
£Zq —— a-^ —— ... —— av ^- u.
Otherwise put
% = Pj (/ = 1,2,3,...) (16)
and
ayj.=(-i)V+/'i; Vir1^
0=1,2,3,...). (17)
Here the formula (17) may also be written in the equivalent form
£ (_iy.-i5Lt»=0. (18)
n= 1 W
Terms of the sum
pj +i - l
£ (-ly1^

An example by Cassels 203
on the right-hand side of (17) can only then be different from 0 if to
their suffix n there exists a positive integer k such that
either j + n = uk or j + n = vk.
This requires in the first case that
l<n = pk + k-j<pj+1-l,
hence
j + l<pk + k<pj+1+j-l
and therefore
j + 1 < pk + k and k < j,
and in the second case that
l<n = pk+1+k-j<pi+1-l,
hence
j+l<pk + 1+k<pj+1+j-l
and therefore
j + l<pk+1+k and k<j-\.
We apply these remarks to the definition (17) of av., but note first
that
(— \y + pk — (— \)p + pk +1 = + 1
for every positive integer k. Hence av. takes the following more explicit
form,
a..=pi+1( £ (-1)'
1 < k < j
pk + k>j + 1
1 i<k<j Pk + k-j
pk + l + k > j + 1
In the lowest case whenj = 1 the first sum has only the term with
k = 1, and the second sum is empty; hence
<*VI = - P2'
On substituting this value, it next follows that
2 2
3/ P P P
av2 = P
/7—1 /r /r — 1

204
The derivative of a function
For larger j the value of aVj becomes even more complicated. It is,
however, still possible to show that always
\aVj\p<p->-1 (/ = 1,2,3,...). (20)
For this estimate is certainly true for j = 1 and j = 2. Assume it has
already been proved for all suffixes 1, 2,... J — 1. Then it remains
valid for the suffix; because by (19),
Uvjlp ^ P
< n j xmax
Pk + k-j
j + i
d + i
+i-j
l<k<j, pk + k>j, 1 </<;-!, pl + 1+l>j).
Here under restrictions on k and I,
\pk + k-j\p>\pk\p and \pl + 1+l
so that the maximum is not larger than 1.
Since by (16),
J\P>\Pl + 1\P,
lim Uj\aUJ\p= lim (p>+j)p J= 1,
j -*■ oo j -*■ 00
and by (19),
lim sup Vj\aVJ\p < lim (pj + l +j)p~j ~ l < 1,
j -*• oo j -*• 00
and since for other suffixes n
«Klp = o,
this proves the assertion (12).
Next, by (18),
00
00
a': =
l (" 1)
n-iaj + n _ V ( l\Pk + k-j-l a"k
n = pj + 1 + 1
72
= I (-1)'
* = j +1
Pk + k-j
00
a
fk
k = j + l p +K J
where 0 <k—j <pk and therefore
\pk + k-j\p = max(\p%,\k-j\p) = \k-j\p>l/(k
and similarly
|pfc+1+/c-jL>l/(/c-j).
-j)

An example by Cassels 205
It follows that
auk
Pk + k-j
<(k—j)p k and
p
Vk
pk + 1+k-j
< (k -J)P
- k
which proves firstly that the two series in the expansion of a,- are
convergent, and secondly that, as j tends to infinity, \a-\p has the
limit 0. Thus both assertions (13) and (14) are true.
Finally the series
has
£ y
k = 0 \K
<■.,! W"
J/ V.
as its first term distinct from 0, while by (16) and (20) all its remaining
terms have at least the factor pj+1. Hence, in the special case when
y=-U
\apj(-l)-(-iypl\p<p-J-\
from which it follows immediately that {an(— l)/n} is not a null
sequence, hence, by Theorem 1, that F'( — 1) does not exist, proving
also the assertion (15).
Theorem 4 among other consequences establishes the fact that even
if the coefficients an of F tend so rapidly to 0 that
lim sup n\an\p= 1, (12)
00
the derivative F' of F still need not exist at all points of Ip, and
similarly for the derivative f'offonj. This is also clear from the two
examples in § 8, for the first function is nowhere differentiable, and pie
second one has a derivative everywhere except at y = — 1.
We shall soon prove a result due to Weisman, as follows. Let a be
an arbitrarily small positive constant. Then there exists a function
F :In-^Qn such that both F and F are continuous on the whole of/n,
P z-'P P>
but that the coefficients an of F tend so slowly to 0 that
lim sup na\an\p= oo.
n -*• oo
We are thus forced to the conclusion that the speed with which an tends
to 0 does not provide a simple law for deciding whether F' exists
everywhere on I or for that matter whether /' does so on J.

206 The derivative of a function
Suppose, however, that the slightly stronger inequality
lim sup n\an\p = 0
n -*• oo
holds. We shall then find that the derivatives /' and F' not only exist,
but are uniformly continuous on J, and on I respectively. In fact, the
stronger property that both functions are strictly differentiable will be
established.
10 Strictly differentiable functions
A function f :J -^ Qp uniformly continuous on J is said to be strictly
differentiable on J if there exists a function g :J x J -^ Qp uniformly
continuous on J x J such that
JW-j(y)= (X ) if XlLy and (x?};)Gjxj. (21)
x-y
Similarly, a function F :Ip->Qp (uniformly) continuous on I is
called strictly differentiable on Ip if there exists a function G:
Ip x Ip-+Qp (uniformly) continuous on Ip x Ip such that
-^ ^-=G(x,y) if x^y and (x,y)elpxlp. (22)
x-y
Let us as usual assume that f is the restriction of F to J. It is then
obvious that g is the restriction of G to J x J,
g(x, y) = G(x, y) if (x, y)eJ xJ.
Theorem 5 If f is strictly differentiable on J, then f exists and is
uniformly continuous on J. Similarly, if F is strictly differentiable
on Ip, then Ff exists and is (uniformly) continuous on Ip.
Proof It will suffice to give the proof for the function F. Denote by y
an arbitrary point of Ip. Then, if xelp tends to y, the limit
p F(x) — F(v) p
FHy) = lim ^ ;_ w = lim G(x, y) = G(y,y)
x -+y X y x-+ y
exists because G is continuous on I x I . This hypothesis implies also
that G(y, y) is continuous on I hence gives the asserted continuity of
F' on Ip.
Let us already stress that the converse of this theorem is not true:

The theorem of Weisman 207
There are functions F which are not strictly differentiable on I but
have the property that both F and F' exist and are (uniformly)
continuous on I \ and similarly with functions f on J.
11 The theorem of Weisman
C. S. Weisman (personal communication 1974) has obtained the
following result which establishes the connection between f or F
being strictly differentiable and the speed with which the coefficients
an tend to 0.
Theorem 6 The function f: J -> Qp is strictly differentiable on
J, and the function F :Ip-^Qp is strictly differentiable on Ip, if and
only if
lim n\an\p = 0. (23)
n -*• oo
Proof Assume first that the function/is strictly differentiable on J,
and denote by
/w-.?/-C)
its interpolation series. The function g in (21) is then uniformly
continuous on J x J and hence, to every positive integer s there
exists a positive integer t = t(s) such that, for elements x, y, X, Y of J,
\g(x,y)-g(X,Y)\p<p-s
whenever
For (x, y)e J x J put
h(x,y) = g(x,x + y+ 1).
Since the inequalities
\x-X\p<p-' and \y-Y\p<p~l
imply that also
|(x + j;+l)-(X+y+l)|p^p-'
and are themselves implied by this inequality together with
\x-X\p<p~\

208
The derivative of a function
it follows that also the function h : J x J -> Qp is uniformly
continuous on J x J.
This means, by Theorem 1 of Chapter 12, that the coefficients cmn of
the interpolation series
x y
HX9y)= ^ Z Cmn]
m = 0 n = 0
X \/y
m \n
satisfy the limit relation
|cmJp->0 as m + 72-xx). (24)
These coefficients cmn can be expressed in terms of the coefficients an
of f. For by the interpolation series for f
kx, y)=m-f^y+J) = i(/(JC+y+1)_ m)
x-(x + y+l) y + l
J
i x + y + 1 r
X
J
and here
x
y + i
k
y + i
-k)\ k
x
J
y
-" + 1).?1iC-*At-i.
because
> + 1\ y + l
k
y
k-\
It follows that
«-*-'%'&%->%'
00 00
-1 ^0^ k\j-
k)\k-l)'
the representation as a double infinite series holds since the binomial
coefficients in the additional terms are equal to 0. For the same reason
the original series for h may be written as
00 00
Hx,y)= Z Z Cmnl
m= 0n = 0
x\fy
m \n

The theorem of Weisman 209
Now, by Chapter 12 § 2, the interpolation series of h is unique. It
follows therefore from the two representations of h that
Jay/fc if j — k = m, k — l = n, where j > k > 1
Cmn (0 otherwise,
or equivalently,
n + I
The limit relation (24) implies then that
->0 as m + ?2-> oo.
am + n+ 1
n+ 1 p
In this formula write AT for m + n + 1 and choose for n + 1 the
largest power pu of /? for which
pu <N and therefore pu+1 > N.
Then
1/AT<|^H"+11,, </VAT.
It follows that
N\aN\p-^0 as JV->oo,
as asserted.
If it is assumed that F is striqtly differentiable on Ip, then the
restriction f of F to J is strictly differentiable on J, and the proof just
given shows again the relation (23).
Finally, let (23) be satisfied. We shall then establish that F is strictly
differentiable on I and therefore f is strictly differentiable on J.
For it was proved in § 2, formula (5), that
F(x + y)-F(y)
= g gn(y) (x - r
or, with a trivial change of variable, that
F(x) - FQQ = g ^W/x-j-r
x-y n = 1 n \ n-\
Here
fc = 0
fe + nj 7 I?

210 The derivative of a function
and therefore
F(x) - F(y)
(25)
< 1 and
p
<1 for x, ye^
p
x-y n=lk = 0 n
In this formula,
K + MP ^ nK + n\P < (k + n)\ak + n\p-^0 as k + rc-> oo.
It follows then by
x - y - 1
n-\
that the double series on the right-hand side of (25) converges
uniformly in (x, y) on Ip x Ip and so defines a function G(x, y) which is
(uniformly) continuous on Ip x Ip, proving the assertion. This
concludes the proof.
12 Pseudo-constants
Before discussing an example of a uniformly continuous function on
J with large coefficients an, it is convenient to introduce the important
notion of a pseudo-constant. This concept opens up a basic
distinction between the analysis of functions of a real variable and that of
functions of a p-adic variable.
In real analysis it is well known that a function which has
everywhere the derivative 0 is a constant. As explained in Chapter 7
§10, this property is a consequence of the real field being ordered by
the relation >. From this derives both the theorem of Rolle and the
mean value theorem of differential calculus.
On the other hand, the p-adic field cannot be ordered (Chapter 7
§ 9), and there are no p-adic analogues to Rolle's theorem or the mean
value theorem of differential calculus. By way of example, the function
F'Ip-^Qp defined by
F(x) = xp-x
vanishes at both x = 0 and x = 1, but its derivative F'(x) = pxp ~ 1 — 1
satisfies
|F(x)+l|p<;l/p if xelp
and so vanishes nowhere in Ip.

Pseudo-constants 111
It is then not surprising that functions on J or Ip may have
derivatives identically zero, without being constant.
Definition A function f :J-^ Qp or F :Ip-^Qp is said to be a
pseudo-constant if
f'(x) = 0 for xeJ, or F'(x) = 0 for xelp,
respectively.
By this definition, all p-adic constants are pseudo-constants. More
interesting are the following two examples.
Firstly assume that f and F are step functions, i.e. that there is a
positive integer t such that
f(*) = f(y) if \x-y\p<p~\ x,yeJ,
or
F(x) = F(y) if \x-y\p<p~\ x,yelp.
It follows then from the definition of the derivative that /' vanishes
everywhere on J and F' does so on Ip. As we may give f and F
different values in the balls
\x-N\p<p'\ where N = 0, 1,..., pl- 1,
these two functions need not be constants, but certainly are pseudo-
constants.
A second example is obtained as follows. Denote by {con} a
sequence of positive integers satisfying
1 < cd1 < cd2 < co3 < ..., lim (con — n) = oo.
n -*• oo
If now xelp has the canonic expansion
2
X =z Xq ~\~ x^p ~\~ X^JP \ • • • 5
where as usual x0, xl9 x2,... are digits 0,1,..., p — 1, define
F:Ip^Qpby
F(x) = x0 + x^' + x2pW2 + . . .
We assert that F is a pseudo-constant on I' .
For let y / x be a second element of Ip with the canonic series
y = yo + yiP + y2P2 + --,

212 The derivative of a function
and let t be the integer for which
\*-y\P = p~\
hence
■*o = 3^0' *i = 3^1? • • • •> xt - l = yt - i> xtzfz yv
It follows that
\F(x)-F(y)\=p
_ n- W(
and therefore
F(x) - F(y)
— jy ~ (Wj - t)
x-y
Now, as x tends to y, t tends to infinity and so does cot — t, whence
F'(y) = 0,
proving the assertion.
The usual rules for differentiation are true for p-adic functions.
They show immediately that the sum, the difference, and the product
of two pseudo-constants on J or Ip are again pseudo-constants; hence
the pseudo-constants on J or on Ip form rings.
If further F : /n -► Q„ and G : In -► In are two functions on /„ with the
P '-'V p p p
derivatives F' and G' and such that at least one of F and G is a pseudo-
constant, then the composite function H :Ip-^Qp defined by
H(x) = F(G(x))
has the derivative
H'(x) = F'(G(x))G'(x) = 0
and so likewise is a pseudo-constant.
Assume the pseudo-constant f or F has the coefficients an\ the
derivative /' or F' naturally has the coefficients a'k = 0. It follows then
from Theorem 2 that
00 n
vn - 1 "* + n
£ (_1)n-lJL±n=() (£ = 0,1,2,...). (26)
n=l n
This homogeneous system of linear equations is thus a necessary
condition for the function f or F to be a pseudo-constant.
It is not known and somewhat doubtful whether it is also a sufficient
condition, nor is the most general null sequence {an} known which satisfies
the equations (26).
It would be of great interest to answer these two open questions.

A pseudo-constant with large coefficients 213
13 An example of a pseudo-constant with large coefficients an
It was already mentioned at the end of §9 that for any a in 0 < a < 1
there exists a continuous function F :Ip-^Qp with a continuous
derivative F' on Ip such that the coefficients an of F satisfy the relation
limsupna|aj = oo.
n -*• oo
An even stronger result, with F being a pseudo-constant on I was
obtained by Weisman (1977) and shall now be constructed. Our proof
will, however, be not the same as that of Weisman.
Denote by {uk} a strictly increasing sequence of positive integers,
1 < u1 < u2 < u3 < ...,
which for the moment may be arbitrary, except that it will already be
assumed that
uk > k + 1 (k = 1, 2, 3,. . .)•
Let Sk =U(p ; uk) be the set of all xelp for which
|x — pk\p<p~Uk, hence also \x\p = p~k.
Thus no two of these sets Sk and Sh where k / /, have a point in
common.
Next let
00
s= U sk
k= 1
be the union of all the sets Sk, and let further &S be the complementary
set of S in Ip,
Denote first by x an arbitrary point of S. There is a unique positive
integer k such that xeSk, and the definition of Sk implies that also
yeSk if \x~y\p<p-uK
Hence S is an open set, as follows of course also from S being the union
of the open sets Sk.
However, the complementary set &S is not open because 0 is the
limit point of the sequence {/?, p2, p3,...} in-S and lies itself in ^S. Let
x be a point of %S distinct from 0.
rf Wp=i,
\y\p = 1 and hence ye*$S
\x - y\p < lip.

214 The derivative of a function
If further
\x\P = P~k and \x ~Pk\P> P~Uk, where /c>l,
then also
\y — Pk\p> P~Uk and hence yeVS if |x - y\p<p~Uk.
It thus follows that every point of^S distinct from 0 is an interior point
of VS.
Define now a function F : Ip -> Qp as follows,
fp2fc if xeSb for some fc > 1,
F(x) = V , .*
[0 otherwise.
By what has just been proved about S and #S, this function is
constant in a certain neighbourhood of every point of I distinct from
0, hence is continuous and has the derivative 0 at such points.
There remains the point 0. Now, as x tends to 0, so does |x|p, and, by
the definition of F, F(x) tends to 0 = F(0), independent of whether x is
in S or in <$S; hence F is continuous at 0. Moreover,
F(x)
X
if xeSk, k>\
0 if xe^S,
p
whence also F'(0) = 0. It follows then that F is continuous and a
pseudo-constant on I
Assume that F has the coefficients an. In order to obtain estimates
for these coefficients, we apply the generating series of Chapter 9 § 5.
Let again z and Z be two indeterminates over Qp related to one
another by the three equivalent equations
z = Z/(l+Z), Z = z/(l-z), (l-z)(l+Z)=l.
Then
OO 00
X F(n)z" = (1 + Z) X a„Z».
n = 0 n = 0
Here the formal power series on the left-hand side can be expressed in
an explicit form by substituting for F(n) its value from the definition
ofF.
We need only consider suffixes neJ which lie in S, say in Sk. Then
F(n) = p2k if \n — pk\p<p~Uk, or equivalently, if n = pk + pUkm where
me J. It follows that
00 00 00 00
X F(n)zn= x p2k Z zpk+pukm= x p2kzpk(\ - zp"kr y
n = 0 k~ 1 m=0 fc = 1

A pseudo-constant with large coefficients 215
On expressing here z in terms of Z it follows that
V 7» l V 2k f Z VY, ( Z V"^"1
00
= £ ^^^(^^^-^-^(I + zk^-z^)"1.
fc= 1
As shown in Chapter 10 §10, the denominator on the right-hand
side can be written in the form
(l + Z)pUk-ZpUk = l-pPk(Z),
where Pk(Z) is a certain polynomial with rational integral coefficients
and of degree pUk — 1. Since
00
(l - Ppk(Z))-* = 2>'i\(Z)',
1 = 0
it follows that
00 00 00
z^n=z y, p2k+lzpk(i+zyuk~pk~lpk(z)1- (27)
n = 0 fc=li=o
On comparing here on both sides the coefficients of Zn, we obtained a
formula for an. We shall do this in particular for the suffix n
= puj — 1 where j is a positive integer, by considering the
contributions to an from the different terms on the right-hand side of (27).
In the term with k = j, I = 0, the coefficient of ZpUj ~ * evidently is
P2j-
Next, all the terms with k >j + 1 and I > 0 have at least the factor
p2j + 2.
There remain the terms
p2k + rZ;>*(1 + Zy»k - pfc - i p^zf wkh l < ^ <j _ l? / > 0,
which are of degree
(pUk -1) + /(p* -1) = (1+ l)(pUk - 1)
in Z and have the factor
n2fc + /
This factor is not greater than p2j if
0 < I < 2(/ - fc).

216 The derivative of a function
Assume therefore that
(l+l)(pUk-l)<pUj-l for 0<l<2(j-k). (28)
Then these terms have too low a degree to make a contribution to the
coefficient of ZpUj ~ 1, and it follows that under this hypothesis
WUJ-% = p-2j 0 = 1,2,3,...). (29)
There remains the problem of satisfying the conditions (28). The
sequence {uk} is so far subject only to the restrictions that ak > k + 1
and uk +1 > uk + 1. Let us instead impose the stronger assumption
that
wx > 2, uk +1 > uk + 2 for k > 1. (30)
We assert that then the inequalities (28) certainly hold.
To show this, we have to prove that
(2j-2k+l)(pUk-l)<pUj-l for l<k<j-l. (31)
This requires for k =j — 1 that
3pUj ~l < pUj + 2,
which is true since by (30)
pUj>p2pUj~1 >3pUj~1.
The trivial inequality
2m + 1 < 3m for m > 2
establishes then that (31) holds also for othqr values of k in
l<k<j-l.
We can now prove Weisman's result.
Theorem 7 Let \l/(n) be a positive integral-valued increasing
function of the variable ne J which tends arbitrarily slowly to oo as
n -> oo. There exists a function F :Ip-^Qp which is continuous and
has continuous derivative F' on Ip such that
lim sup \l/(n)\an\p = oo. (32)
n -*• oo
Moreover, F may be chosen as a pseudo-constant.
Proof The sequence {uk} could so far be any sequence with the
properties (30). Assume now that it increases so rapidly that
lim ijj(pUj — \)p~ 2j = oo.
j-+ °°

The rules for differentiation 111
If F is the function obtained by the last construction, the assertion (32)
is a consequence of the equations (29).
The function F, as constructed, is a pseudo-constant. As we may
add to F a polynomial, so changing at most finitely many coefficients
an, there are also functions with derivative not identically 0 and
continuous on Ip with the property (32).
It is obvious from Theorem 6 together with Theorem 7 that a
function F may have a continuous derivative on Ip without being strictly
differentiate.
14 The rules for differentiation
The well-known rules from real analysis on the derivatives of a sum,
difference, product, or quotient of two functions, or on the composite
of two functions, have their analogues for functions of a p-adic
variable on J or on I Moreover, the proofs are essentially the same,
except that the real limits must be replaced by p-adic limits.
By way of example, let F :Ip-^Qp and/G : Ip -► Qp be two functions
which at some point y of Ip have the derivatives F'(y) and G'(y) ,
respectively. Then
(F(x + y)± G(x + jO) - (F(y)± G(y))
= (F(x + y) - F(y)) ± (G(x + y)- G(y))9
whence, on dividing by x ^ 0 and allowing |x[ to tend to 0, it follows
that
the derivative of F ±G at y is F'(y) ± G'(y). (33)
Next,
F{x + y)G(x + y)-F{y)Giy)
= (F(x + y)-F(x))G(x + y) + F(x)(G(x + y)-G(y)\
so that, by the same procedure,
the derivative of FG at y is F'(y)G(y) + F(y)G'(y). (34)
Thirdly, on assuming that G(y) ^ 0 and therefore, by the continuity of
G at y9 also G(x + y) ^= 0 for all sufficiently small |x|p,
F(x + y) F(y) = (F(x + y)- F(y))G(y) - F(y)(G(x + y)- G(y))
G(x + y) G{y) G(x + y)G(y)

218 The derivative of a function
the derivative ofF/G at y is L..2 — (35)
from which it follows in the same way as before that
F'(y)G(y)-F(y)G'{y)
G(y):
Analogous formulae hold for functions f and g on J and are proved
in the same way.
Also the derivative of a composite function H = F(G) is given by the
same rule as in real analysis. Let F:In-^Qn and G :/„-►/„ be
j p y^p p p
differentiable at the points G(y) and y, respectively; note that G is
assumed to map Ip into itself in order that the composite function
H(x) = F(G(x)) for xelp
is defined. We distinguish two cases.
Firstly, there may exist an infinite sequence {x(n)} of distinct points
of Ip with the limit y such that
G(x<">) = GOO (n = 1, 2, 3,...),
and therefore also
H(x<">) = HO0 (72=1,2,3,...).
By the existence of G'(y) this can only happen if G'{y) = 0, and if H is
differentiable at y, it follows that also H'(y) = 0.
If, secondly, x tends to y such that always G(x) ^ G(y\ then
H(x) - H(y) = F(G(x)) - F(G(y)) G(x) - Gjy)
x-y G(x)-G(y) x-y
and here, by the assumed existence of F'(G(y)) and G'(y), the right-
hand side has the limit F'(G(y))G'{y) as x tends to y. This limit is 0 if
G'(y) = 0? Just as we found for the exceptional sequences {x{n)} in the
first case. Hence the following result holds,
the derivative of F(G) at y is F'(G(y))G'{y). (36)
An analogous formula holds for functions f :J -^ Qp and g:J-^J.
15 Problems
1 Let m be a positive integer. Determine the interpolation series of the
( x
derivative F'(x) where F(x) =
\m
2 Which conditions must the p-adic sequence {cn} satisfy if the

Problems 219
function F :I -^Q defined by
' x
FW=.?oHp"
is to be continuous and differentiable on II
3 Discuss the differentiability of
00
F(x) = ^ t
n=0
at all points of Ip.
4 Let F :In-^Qn be the function
P *^P
1 ; \l/\x\p if x^O.
Is F (i) locally constant, and (ii) a pseudo-constant on /p? Give a
sequence of pseudo-constants that converges on I uniformly to F.
5 Let x = x0 + xx p + x2p2 + ... be the canonic series of xe J and let
c.J->Qp be the function c(x) = x0. Characterise all functions
/ : J -> Qp for which c(x)/(x) is a pseudo-constant.
6 Let F :/p-►Qp be the function F(x) = £ nlf * J. Show that F'(x)
is a p-adic integer for every xel .

14
Higher derivatives
1 The iffth derivative
Let, as in § 1 of the last chapter, f : J -► Qp be uniformly continuous on
J and F :Ip-^Qp continuous on lp\ assume, moreover, that f is the
restriction of F to J. Both functions have the same coefficients a
where naturally
n
P
lim an = 0,
«-*• 00
and
/(*)= Z M J and ^W= ian\
If the derivatives/' of/and F' of F exist on J or Ip9 respectively,^'
evidently is the restriction of F' to J. If, moreover, /' is uniformly
continuous on J and F' is continuous on Ip, then they have the same
interpolation coefficients a'n where
and these satisfy
and
00 a
<= E (-l)*"1-^ (n = 0,1,2,...), (1)
k= 1 ^
lim ^ = 0,
n -*• oo
xx\ " /x
/'(*) = Z < and *"(*) = E a,,
n = 0 \nJ n = 0 \n,
The functions /' and F' may themselves be differentiable, with the
derivatives /" and F". If /" is uniformly continuous on J and F" is
continuous on I then /" is the restriction of F" to J, both functions
have the same coefficients a'n' where
a
a
';= £ (-l)'-i^ („ = 0, 1,2,...), (2)
k=l k

A function with many derivatives 221
and
and further
lim a!; = 0,
«-*• 00
x\ " /x
/»(x)= x< and *""(*) = £ <,
n=0 \n/ n=0 \ny
If this process of differentiation can be repeated m times, we obtain
the mth derivatives f(m) of/ and F(m) of F; again /(m) is the restriction
of F{m) to J. Assume in particular that /(m) is uniformly continuous on
J, and F(m) is continuous on I Both functions have the same
coefficients, a£m) say, and
lim aim) = 0.
n
n -*• oo
Further
x\ . _, £ , Jx
m)
fW(X)= X ^M and ^m)M= Z «2
n = 0 \n/ n=0 \n
for x on J and / respectively. Recursively in m,
4m)= Z (-l)*"1-^ (11 = 0,1,¾...). (3)
2 A function with many derivatives may still have large
coefficients
The sequence of coefficients {an} defines both / and F uniquely.
Hence properties of these functions, like that of having m continuous
derivatives, should find some expression in properties of this
sequence. However, such a property must be rather deep and cannot
simply be expressed by the speed with which the coefficients tend to 0.
This is clear from the following theorem.
Theorem 1 Let ^(n) be a positive integral-valued increasing
function of ne J which tends arbitrarily slowly to oo as /r->oo.
Then there exists a function F : / -» Qp with continuous derivatives

222 Higher derivatives
of all orders on Ip such that
limsup^(fi)|an| = oo.
«-*• 00
Proof The function F constructed in the proof of Theorem 7 of
Chapter 13 §13 has the required properties. For since it is a pseudo-
constant, all its successive derivatives are 0 on Ip and so are
continuous, just as F itself is. However, its coefficients have the
asserted property.
3 A function with very small coefficients has many continuous
derivatives
As a converse to Theorem 1 we show now that small coefficients do
imply that the functions f and F have many continuous derivatives.
Theorem 2 Let mbea positive integer, and let the coefficients an of
f :J-+Qp and F \Ip-^Qp satisfy the relation
lim nm + 1|a„L = 0. (4)
"*n\p
n -*• oo
Then f /',..., f(m) are uniformly continuous on J, and F,
F\ . . ., F(m) are continuous on Ip.
Proof By (1) and (4),
nm\a'n\p< supn™
k>l
ak + n
< SUp/CT2mK + n|
k> 1
< sup(k + ri)m+1\ak + n\p-^0 as n-^oo.
k> 1
On repeating this procedure with (2) and (3) instead of (1) it follows
successively that
rF-^a'^p^O, nm_2|ai3)|p^0,..., n\a^\p-^0 as n-»oo.
From these formulae, the assertion follows by Theorem 6 of Chapter
13. In fact, we obtain the stronger result that /,/',..., f{m) are strictly
differentiable on J,andF,F',..., F(m) are strictly differentiable on Ip.
Theorem 2 immediately implies the following result.

Analytic functions on J and on Ip 223
Theorem 3 Let \jj(n) be a positive integral-valued function of
neJ which increases to infinity as n tends to infinity. If
lim n*in)\an\p = 0, (5)
n -*• oo
then both f and F have continuous derivatives of all orders on J
and on Ip, respectively.
Proof. The relation (5) implies the relation (4) for every integer m.
4 Analytic functions on J and on Ip
If f : J -^ Qp or F :Ip-^Qp have continuous derivatives of all orders,
we may form their Taylor expansions
oo y.m oo vm
I /(m)(0)— and £ i*»>(0)- .
and ask whether these series converge and represent the functions.
This is analogous to what one does in the cases of functions of a real or
complex variable..
Unfortunately, this Taylor series will not in general converge on J
or on I and even if it does converge, it need not represent f on J or F
on I A trivial example is given by any non-constant pseudo-
constant; its Taylor series reduces to its first term and is a constant
We therefore introduce the following notation.
An infinite series of the form
00
Z AX
n = 0
with coefficients An in Qp is called a power series over Q A function
which on J or on Ip is equal to a convergent power series over Qp is said
to be analytic.
Assume, say, that F :Ip-^Qp is equal to the convergent series
00
F(x)= X Anxn (6)
n = 0
for all xelp. Then the power series converges in particular at x = 1,
and hence
p
lim An = 0.
n -*• oo

224 Higher derivatives
Conversely, if this limit condition is satisfied, then the power series
converges everywhere on Ip and so also on J, and the convergence is
uniform in x. Hence the analytic function F(x) is continuous on I
and its restriction f(x) to J is uniformly continuous on J.
It follows that F (and similarly f) can be written as a convergent
interpolation series
fW = ioa"(n} ™
There arises then the following problem.
Under which conditions on the coefficients an does this interpolation
series represent an analytic function on Ip or on J ?
The answer is quite simple and as follows.
Theorem 4 The function f :J -^ Qp and the function F :Ip-+Q
are analytic on J, and on Ip, respectively, if and only if
p a
lim ^ = 0.
n -> oo
n\
Proof It is easily established that there are rational integers S(n, k)
and s(n, k) such that identically in x,
xn= Y S(n,k)kl(*) and n!| )= Y s(n, k)xk
k = o W W k = 0
(.2 = 0,1,2,...), (8)
where, in particular,
S(n, n) = s(n, n) = 1.
For both xn and n\\ ) are polynomials in x of the exact degree n, with
w
highest coefficients 1 and with rational integral coefficients.
Assume now, firstly, that F can be written as the convergent power
series (6). Then, by the first formula (8),
oo n ^y\ °° / Y
F(x)= £ An X S{n,k)k\ = £a
n=0 k=0 \K/ k=0
/C / u — c\ \ K
and on comparing the coefficients of [ ) on both sides of this

Analytic functions on J and on Ip 225
equation, it follows that
a
k\
k- = £ AHS(n9 k) (k = 0, 1, 2,...).
n = k
Here | S(n, k) \p < 1 for all k and n, while An has the p-adic limit 0; hence
the same is true for the sequence {ajk\}.
Secondly, assume that the interpolation coefficients have the
property that {ajn!} is a null sequence. Then, by (7) and the second
formula (8),
F{x)= £ tAlr ^t^ s{n'k)xk= s a**,
n=0n\ \nj n=on-k=o k=o
whence, on comparing the coefficients of xk on both sides of this
equation,
A= t-,s(n,k) (k = 0, 1, 2,...).
n = k'L '
Using now that \s(n, k)\p < 1, it follows that {Ak} is a null sequence.
In Theorem 4, the p-adic value of n! can be expressed explicitly. For
this purpose let n have the canonic form
n = n0 -\-nlp-\- n2p2 + ... + nrp\
where n0, nl9 n2,..., nr are digits 0, 1,..., p — 1, and where nr > 0.
(The trivial case when n = 0 and 0! = 1 has been excluded.) Put
{n} = n0 + nr + n2 + .. . + ftr.
Of the positive integers not exceeding n exactly
n1 +n2p + .. . + nrpr~ 1
are divisible by at least p1; exactly
n2 + n3p + . .. + nrpr~2
are divisible by at least p2; etc.; exactly
nr_x+nrp
are divisible by at least pr ~ x; and finally exactly
nr
are divisible by p\ while none are divisible by pr + 1.
Let p" denote the largest power of p that divides n\. Evidently,
u = (n1+n2p + ... + nyP^1) + {n2 + n3p + ... + 7ir//"2)
+ (w3 + n4p + ... + nrpr ~ 3) + . • • + (nr _ 1 + nrp) + rcr

226 Higher derivatives
Combine here all the terms which have the same digit nk as a factor,
and use the formula for the sum of a finite geometric progression. It
follows that
p-1 p2-l p3-l , f~l-\ , f~\
u = n1 T + n2 - + n3 - + .. . + nr_ x \-n
p-1 p-1 P-1 r p-1 r p-1
which is equal to
1 n — {n}
u = -(nn + 7ii p + . .. + nrpr — (n0 + n, + . .. + nr)) = -—.
p-lu 1 p-1
Hence
\n\\p = p-{n-{n))l{p-l\ (9)
The condition of Theorem 4 is therefore equivalent to
lim p(n-^^-^1^1- = 0. (10)
n -*• oo
5 Two elementary /i-adic functions
There is a well-developed theory of analytic p-adic functions which,
however, shall not be discussed. Instead, we shall study two basic
elementary analytic functions from our theory of interpolation series.
Let a be any p-adic number satisfying
\a—l\p< 1/p and therefore \a\p = 1. (11)
We firstly define a function F : Ip -► Qp by the interpolation series
F(x;a)= £ (a-l)"fXY (12)
n=0 \n/
Since its coefficients an = (a — l)n tend to 0 as n tends to infinity,
F(x; a) is a continuous function on Ip9 and its restriction f(x; a) to J is
uniformly continuous on J. By the binomial theorem,
f(x; a)= £ (a - l)n(X) = (1 + (a - l))x = of.
n = 0 W
Next let x, yelp. Then
«»M= iak + P\= ft(a-lf + "(y)=(a-lfF(y;a),
k = 0 \K/ k = 0 \K

Two elementary p-adic functions 227
so that by the general addition formula (4) of Chapter 13 § 2,
00 /y \ 00 I y.
F(x + y;a)= £ an(y)[ = I (a-l)"F()>;a)
n = 0 \W/ n = 0 \n
that is,
F(x + y;a) = F(x;a)F();;a) for x,yelp. (13)
On restricting x and y to J, this identity reduces to the equation
ax + y = axay (13*)
which is obvious from the associative law in Qp.
Conversely, (13) follows from (13*). For let {xn} and {yn} be two
sequences in J with the limits x and y, respectively. Since F is
continuous on Ip, the three sequences {aXn + yn}, {aXn}, and {ayn} have
the limits F(x + y;a), F(x;a), and F(y;a), respectively, and since
aXn + yn=aXnayn, we again obtain (13).
Denote by b a second p-adic number satisfying
\b-l\p<l/p.
Then, by the same kind of proof, we find that also
F(x;ab) = F(x;a)F(x;b) for x,yelp. (14)
For when x is restricted to J, this is
(ab)x = axbx, (14*)
an equation which follows from the commutative and associative
laws in Qp, and from which (14) can again be derived by a simple
limiting process.
Let again y be any element of Ip. Not only
{an(y)} = {(a-l)nF(y;a)},
but also
is a null sequence; for
lim np ~n = 0.
«-*• 00
Hence the derivative
F'(y;a)= £ (-1)-^ = F(y;a) £ (-1)-1^^
n= 1 W n= 1 w

228 Higher derivatives
exists. Let us write L(a) for the convergent series
00 (a - If
L(a)= X(-l)-1^—L. (15)
n= 1 U
On writing again x for y9 we have then the differentiation formula
F'(x;a) = F{x;a)L{a), (16)
hence for every positive integer m,
F(m)(x; a) = F(x; a)L(a)m. (17)
7¾ expression L(a) defines the p-adic logarithm. If also \b — 1|
< 1/p, then £/ze functional equation
L(ab) = L(a) + L(b) (18)
Zio/ds. It follows immediately on differentiating the identity (14) with
respect to x, noting that for x = 0 both sides are equal to 1, and thus
do not vanish. For by the law for the derivatives of a product, this
gives
F'(x; ab) = F'(x\ a)F(x\ b) + F(x; a)F'{x\ b),
that is,
F(x; ab)L(ab) = F(x; a)F(x; b)L{a) + F(x; a)F(x; b)L{b\
which is (18).
The coefficients an = (a— If of F(x; a) have the property
lim Pn\an\p<h
n -*■ oo
as follows from (11). Hence, for p > 3, the condition (10) is satisfied; for
{n} is of smaller order than n for large n (it is at most of order log n). It
follows therefore from Theorem 4 that F(x; a) is analytic for p>3.
On the other hand, if p = 2 and \a— l|2=i, then F(x\a) is not
analytic, although it has derivatives of all orders. For now
lim 2X1, = .1,
«-*• 00
while the condition for analyticity (10) requires that
lim 2B-{B}|aJ,=0.
ln\2
n -*■ oo
However, when n = 2k is a positive integral power of 2, then
W = {2*} = 1, 2"-'»>|a„|2=i
and so this condition is not satisfied.

ax when a does not satisfy \a — 1| < 1 229
However, if the stronger inequality
\a-\\2<k
holds, then F(x;a) is again analytic on I because this hypothesis
evidently implies that (10) is true.
But let us return to the case when \a — 1|2 — 2. From the functional
equations (13) and (14), it follows that
F(2x; a) = F(x; a)2 = F(x; a2).
Here \a2 — 112 < i, and hence F(x; a2) is analytic on I2; this identity
implies then that F(x; a\ while not analytic on I2, can be written as a
convergent power series on the smaller set |x|2<|. By way of
example, the function F(x; — 1) is a pseudo-constant on /2, being
equal to + 1 if |x|2 <i and equal to — 1 if |x — 1|2 <\.
In the 2-adic case, L(— 1) is defined,
oo On
L(-l)=- I-'
Since
2L(-1) = L(1) = 0,
L(— 1) also is equal to 0, a somewhat surprising fact.
To return to the case of general p, it is clear that L(l +px) is
analytic on I Its coefficients, ln say, have the values
h=io(-v"~k(t)L(i+kp).
It follows therefoie from (18) and from Theorem 4 that
n
lim p(n ~{n})/(p ~ 1]
n -*■ oo
l( n (i+M(_i)n"fc®
vfe .= 0
=,0.
This equation remains true if the function sign L is omitted, and a 1
subtracted from the product instead.
6 The function ax when a does not satisfy | a — 1 \p <1
The function F(x; a) which was the extension of f(x; a) = ax to I was
defined under the restriction that
\a—1\ <l/p, hence \a\ = 1.

230 Higher derivatives
Assume now that a is a p-adic integer which does not satisfy
\a — l\p < 1/p. Then/(x; a) = ax still has a good meaning for xg J and
can be written as the interpolation series
f(x;a) = £ (a-l)»(X\
n = 0 W
However, now the coefficients (a — l)n do not tend to 0, and hence
f(x; a) is not uniformly continuous on J and so cannot be extended
to a continuous function on Ip. In particular, if \a\p < 1, then f(x\a) is
discontinuous at every point x of J (compare the second example in
Chapter 8 § 4).
Let, however \a\ = 1. Then aelp can be written in the form
a = a0 + bp,
where a0 is one of the digits 1, 2,..., p — 1, and b lies in Ip. By the
binomial theorem,
and here, by Fermat's theorem, the rational integer
is divisible by p. It follows that av ~ x has the form
ap~1 = l + pB,
where Belp. In .other words,
la'^-llp^l/p,
and therefore /(x;ap_1) is uniformly continuous on J, and its
extension F(x;ap~ x) is continuous on I
If xe J, it has a unique representation
x = (p-l)y + z9 (19)
where y lies in J and z is one of the numbers 0,1,..., p — 2.
Correspondingly, we may define f(x;a) as
f(x;a) = (a*-1yaz = azfty;a*>-1\
and here az assumes only the p — 1 values
1,a,a2,... ,ap~2. (20)
The factor f{y\ap~ l) is again uniformly continuous for ye J.

The exponential series 231
Assume next that x lies on Ip. Then, for each of the values (20) of az,
there is a unique yelp such that (19) is satisfied. Hence, if we try to
define F(x;a) by the formula
F(x;a) = azF(y;ap-1),
then F(x; a) becomes a multivalued function of x and so falls outside
our considerations.
One therefore deals not with F(x\a\ but with
F({p-\)x\a) = F(x\a*-%
so obtaining a function which is continuous on I and given by the
series
00
F((p-l)x;a)= £ (0^-1)
n = 0
If p > 3, this function is again analytic on Ip. If, however, p = 2, then
the two cases discussed in the last section have to be distinguished.
7 The exponential series
In real or complex analysis, the exponential function qx is defined by
oo -^n
where e is defined by the series
00 I
n = On '
In the p-adic case, these series are no longer convergent if |x| < 1 and
in particular if x = 1. If, however, the function epx for p > 3 and the
function e4x for p = 2 are considered, then we are again dealing with
functions analytic on I In particular, ep for p > 3 and e4 for p = 2 lie
in I A number analogous to e lies in an algebraic extension of Qp, of
degree p if p > 3 and of degree 4 if p = 2.
It can be shown that also in the p-adic case the p-adic exponential
function is inverse to the L-function, i.e. that
eL(x) = L(ex) = x,
provided that x is a p-adic number for which the functions in this
:

232 Higher derivatives
formula are defined. This result belongs to the theory of p-adic
analytic functions and so lies outside our considerations.
8 Problems
1 The continuous function F : I -> Qp is defined by its interpolation
series
00 (x
n = 0 \n,
and has a continuous second derivative with the interpolation
series
00 Ix
n = 0 \n,
Determine the coefficients a'n' in terms of the coefficients an.
2 P0(x), Pi(x\ ..., Pp_ i(x) are p polynomials in x with coefficients
in Q A function F :Ip-+Qp is defined by
F(x) = PX0(x),
where x0 denotes the first digit in the canonic series x = x0
+ xtp + x2p2 + ... of x. Determine the successive derivatives of
F.
3 Cl\Ip-^Qp and C2 : /p -► Qp are two pseudo-constants on Ip where
|Cx(x)- l\p< l/p and |C2(x)|p^l for xelp.
Determine the successive derivatives of the function F :Ip-^Qp
where
00
F(x)= £ C1(x)"C2(x)x-.
n = 0
4 F : /p -► Qp is the function
F(x) = I (n + [v^])!
where [^n] denotes the largest integer not greater than^A. Decide
whether F is analytic on I
5 Let a be a p-adic number satisfying \a — 1 \p < l/p, and let F :Ip-^Qp
and G : I -> Qp be the two functions defined by the series
oo / X \ °° ( X
F(*)- Eo(-l)"U and G(x)= Zo(a-ir(2n + i

Problems yxT.
Express F(x + y) and G(x + y) in terms of F(x), G(x),F(y), and G(y),
and determine coefficients A, B, C, and D that depend only on a
such that
F'(x) = AF(x) + BG(x) and G'{x) = CF(x) + DG(x).
6 Prove that the function F :Ip-^Qp defined by
has on Ip derivatives of all orders. Is this function analytic on I ?

15
The Dieudonne integral
The following notations will be used in this chapter.
The letters c, and C, will denote pseudo-constants on J, and on I
respectively; thus the functions c:J-^Qp and C :Ip-^Qp have the
properties
c'(x) = 0 for xeJ, and C'(x) = 0 for xelp,
respectively. We know that there are many pseudo-constants which
are not constants, and we shall in due course establish explicit
formulae defining classes of such pseudo-constants.
Next denote by g :J -+Qp an arbitrary function on J, and by
G : Ip -> Qp a continuous function on Ip. An (indefinite) integral/of g is
any function/ :J -+Qp such that
f'(x) = g(x) for xeJ,
and similarly an (indefinite) integral F of G is any function F :Ip-^Qp
with the property that
F'(x) = G(x) for xelp.
It is obvious that with f also f + c is an integral of g, and similarly
with F also F + C is an integral of G. There are thus large classes of
integrals of both g and G. In these classes we shall select one element
each with particularly simple properties and call the integrals so
defined the Dieudonne integrals of g and of G, respectively. Actually
Dieudonne (1944) solved the more general problem of the solutions of
a first-order differential equation and defined for each such equation
an infinite sequence of his integrals.
1 Definition of the Dieudonne integral of g
Write the variable xe J as a canonic series
x = x0 + XtP + x2p2 + ...,

Consequences of the boundedness of g 235
where the coefficients xn are digits 0, 1,..., p — 1. Since x is a non-
negative rational integer, the series breaks off after finitely many
terms, and hence there exists a suffix r(x) > 0 such that
xn = 0 if n > r(x).
Put
x(n) = x0 + xx/7 + ... + xn _ xpn 1 (n = 1, 2, 3,...),
so that
x(n + 1) _ x(n) = ^pn = q if „ > r(x).
Analogous notations will soon be used for a number y in J or in /p.
The Dieudonnt integral f(x) of g(x) is now defined as the series
00
f(X) = £ (X<» + !» - X(">)0(X(">). (1)
n= 1
Since the terms of this series with n > r(x) are equal to 0, there is no
problem of convergence.
By way of example, the function g(x) = 1 has the integral
00
f(x) = X (*(n + 1} - *(n)) = *(r(n)) - *(1) = x - X0,
n= 1
and the function g(x) = x has the integral
00 y.2
f(x)= Y, (x(n + 1] - x(n))x(n) =--+ h(x\
n= 1 2
where by simple algebra
h(x) = - \{x20 + x\p2 + x\pA + ...). (2)
It will soon be proved that h(x) is a pseudo-constant.
The behaviour of the Dieudonne integral f of g will now be studied
under suitable assumptions for g.
2 Consequences of the boundedness of g
The function g could so far be arbitrary. It is clear from the definition
that the Dieudonne integral f always satisfies the inequality
sup|/(x)|p<sup|0(x)|p; (3)
xej xej

236 The Dieudonne integral
for the factors
x(n + 1) _ x(n)
in (1) have at most the p-adic values 1.
Let y be a fixed number in J. The function g is said to be bounded at
y if there exist to y two positive integers t and u such that
\g(x)\p<pu if \x-y\p<p~\ (4)
This property has the following consequence.
Theorem 1 If g is bounded at y, then its Dieudonne integral f is
continuous at y.
Proof Let s be an arbitrarily large positive integer. The hypothesis (4)
remains satisfied if t is replaced by any larger positive integer; it may
therefore without loss of generality be assumed that
t > s + u and t > 2.
Let x be an arbitrary element of J satisfying
\x-y\p<p-\ (5)
Then in the canonic series for x and y9
xn = yn if 0 < n < t — 1
and hence also
x(n) = y(n) jf ^ < n < t
It follows that
00
/M - /GO = Z ((*<"+ J) - x<-W>) - (/•+ « - y("»)^(3;<")))
n= 1
oo
= I ((*(n + 1} ~ x(n))g(x(n)) - (y(n +1)- y(n))g(y(n))).
n = t
In the second sum the suffix n is at least t, so that
\x(n+1)-x(n)\p<p-n<p-\ \y(n+1)-y(n)\p<p-n<p-\
and also
Wn\-x\p<p-n<p~\ lyW-yl.^p-^p-'.
Further by (5),
\x^-y\p<\(x^-x) + (x-y)\p<m^x(\xin)-x\p^x-y\p)<p-t.

Consequences of the continuity of g 237
The hypothesis (4) implies then that
\g{x*%<p\ \g(y(%<p»,
whence
\f(x) - f(y)\P < sup(|(x<»+ » - x^)g(xin%,\(/" + " - y*W%)
n> t
<P~tpU<P~\
giving the assertion since s may be chosen arbitrarily large.
When g is bounded on J, the integer u is independent of x and y.
Hence the following result holds.
Theorem 2 If g is bounded on J, then the Dieudonnk integral f is
uniformly continuous on J.
3 Consequence of the continuity of g
The next theorem is analogous to a classical property of the Riemann
integral in real analysis.
Theorem 3 If g is continuous at y, then the derivative f'iy) of the
Dieudonne integral exists and is equal to g(y).
Proof By the hypothesis there exists to every positive integer s a
second positive integer t = t(s, y) independent of x such that
\g(*)-g(y)\P<p~s if \x-y\p<p-\ (6)
Hence, in particular, g is bounded on the set (5), and an inequality (4)
holds.
It suffices to consider numbers xeJ which are distinct from y so
that
\x-y\P = p~T,
where the rational integer T is at least equal to t. This means that
xn = yn if 0<n<T-l and x(n) = y(n) if l<n<T.
Hence, just as in the proof of Theorem 1,
oo
/M-/00= I ((x("+1>-x<»>)0(x»">)-(y"+i>-/"^(y"))), (7)
n = T

238 The Dieudonne integral
where again only finitely many terms of the infinite series are distinct
from 0.
Also just as in this proof, for n > T,
|x(n + 1)-x(%<p-T, \y(n + 1)-y(n)\p<p-T,
\x^-x\p<p~T, \y{n)-y\p<p~\ \x^-y\p<p~^
whence, by (6),
\g(x^)-g(y)\p<p-s and \g{y{n))-g{y)\p<p-s.
Therefore write
g(xW) = g(y) + gn and g(y(n)) = g(y) + Gn;
then
\gn\p<p~s and \Gn\p<p~s for n>T.
By the equation (7),
00
f(x)-f{y) = g(y) £ ((*<"+1] - x(n)) - (y{n + *> - /">))
n= T
oo
n= T
Since x(T) = y(T), the first series is equal to
(x _ ^)) _ (y _ yT)} = x _ ^
while the p-adic value of the second series does not exceed
sup (i(x(»+x> - x<»>)0„ip, i(y»+» - yn))G„iP) < P- Vs
n>T
= \x-y\Pp~\
and it follows that
/(*) - /GO .,
--giy)
<p~s- (8)
P
x-y
Since s may be made arbitrarily large by choosing t sufficiently large,
this gives the assertion
x-+ y x y
Theorem 4 If g is continuous at every point of J, then the
Dieudonne integral f has the derivative f'(y) = g(y) at every point
y of J.

An example 239
By way of example, the Dieudonne integral of g(x) = x was equal to
\x2 + h(x) where h(x) was the function (2). Here g is continuous on J,
and the derivative of \x2 is equal to x = g(x). Therefore on J,
h'(y) = 09
showing that h is a pseudo-constant on J, as was stated already in § 1.
4 An example
We show next, by means of an example, that a function g may be
everywhere discontinuous on J and still have a Dieudonne integral
which, moreover, is continuous and differentiable at all points of J.
Put g(x) = Px for xeJ.
The sequence
{P°, v\ V\ • • •}
is a null sequence in which all terms are distinct from zero. This
implies by Chapter 8 § 4, that g is discontinuous at every point y of J.
For, firstly, g(y) ^ 0. Secondly, as x ^ y tends to y9 x runs over positive
rational integers that become arbitrarily large, and so
p
\img(x) = 0*g(y).
x-+y
The Dieudonne integral f of g is equal to
00
f(x)= X (xin+1)-x(n))px
n= 1
or more explicitly,
/(x) = xypl + Xo + x2p2 + Xo + X1P + x3p3 + Xo + X1P + X2p2
_|_ _|_ v rf + x° + XlP + X2p2 + ' ' • + x" - iPn ~ 1 +
Let now x and y be two distinct numbers in J, say, such that
\x -y\P = p~T-
Then
xn = yn for n = 0,1,..., T — 1.
Therefore by the last equation,
f(x) - f{y) = (xT - yT)pT + X0 + X1P + X2p2 + • • • + *r- iPr- *
+ terms in higher powers of p.

240 The Dieudonne integral
It follows that
/M - /GO
< n_ (*o + Xip + x2p2 + . . . + xT_ 1pT - l)
x-y
As x, while remaining always distinct from the fixed y, tends to y9 at
least one digit xn of x with arbitrarily large suffix n is distinct from 0,
and so the right-hand side of this inequality tends to zero.
Therefore
&mzm=r(y)=0 for yeJt
x -*■ y x y
which proves that f is differentiable at every point of J and is a
pseudo-constant. Naturally f'^g. The Dieudonne integral of a
discontinuous function need not be an integral!
5 Two further examples
The two functions
0i(x) = (x + l)_1 and g2(x) = (x+ l)"2
are bounded and continuous at every point of J; their Dieudonne
integrals /t (x) and f2 (x) are therefore continuous and differentiable at
every point y of J, and
f i iy) = G i GO and f2 (y) = g2 (y)
if ye J. On the other hand, neither of gy and g2 is bounded on J, both
assuming arbitrarily large values as y tends to — 1.
Let y run over the numbers
y = p"-l=(p-l) + (p-l)p + ... + (p-l)p"_1 (11 = 1,¾¾...).
A simple calculation gives
/i(p" -1) = i (p - i)p*-(p*rx=»(p -1),
* = l
and
/2(p"-i)= Z (p-i)pk(pkr2 = i-p-n-
k = i
Hence, as n tends to infinity, the sequence {fi(pn— 1)} remains
bounded, but has no p-adic limit, while the sequence {f2(pn — 1)} is

The Dieudonne integral ofG 241
not bounded. Thus neither /x nor f2 can be extended as continuous
functions into the point — 1 which does not belong to J.
6 Definition of the Dieudonne integral of G
Next let G :Ip^>Qp be a continuous function on Ip. Its restriction
g : J -> Qp to J defined by
g(x) = G(x) for xeJ
is therefore uniformly continuous.
Write the variable xelp again as a canonic series
x == Xq "t~ X-±P t X2P ' * * • •'
and put
x(n) = x0 + xx/7 +... + xn _ x/7n" l (n = 1, 2, 3,...).
Now the series for x need not terminate after finitely many terms.
Put
n- 1
Fn(x)= £ (x(k + 1)-x(k))G(xik)) + (x-x(n))G(xin))
(n=l,2,3,...X (9)
and similarly,
n
FB + 1(x)= X (x(*+1)-x(*))G(x(*)) + (x-x(" + 1))G(x(" + 1)).
k= 1
On subtracting these two equations,
F„ + 1(x)- Fn(x) = (x - x<"+ ^)(0(^-+ ") - G(x<">)). (10)
In this formula,
\x-x(n + 1)\p<p-(n + 1\ (11)
By hypothesis, G is continuous on /p. Hence there exists to every
positive integer s a second positive integer t = t(s) independent of x
and y such that
\G(x)-G(y)\p<p-s if x,yelp and |x - y\p </?"'. (12)
Since
|x(n+1)-X(n)|p</7"n,
it follows that
|G(x(n + 1))-G(x(n))|p</7"s if n>t,

242 The Dieudonne integral
hence, by (10) and (11), that
\Fn + 1(x)-Fn(x)\p<p-(" + »-s if n>t. (13)
On applying this formula for the suffixes n, n + 1,..., m — 1 and
adding, we find that
\Fm(x)-Fn(x)\p<p^n+1)-s if m>n>U (14)
uniformly in x.
Hence {Fn(x)} is a fundamental sequence, and so the p-adic limit
p 00
lira Fn(x)= ^ (x(k + 1)- xw)G(x(k)) = F(x), say (15)
n->oo k = 1
exists. Moreover, the functions Fn(x) are by their definition
continuous on Ip, and therefore F(x) is a continuous function on Ip.
Next, on allowing m to tend to infinity in (14), it follows that
\F(x)-Fn(x)\p<p-("+v-s if n>t (16)
The function F(x) so defined is again called the Dieudonne integral of
G. If f and g denote the restrictions of F and G to J,
f(x) = F (x) and g (x) = G (x) for x e J,
then, by (1) and (15),/ is also the Dieudonne integral of g according to
the definition in § 1.
7 The Dieudonne integral F of G is strictly different able
Denote by x and y two distinct elements of Ip and define the rational
integer T by
I* - y\P = p~T-
Let s and £ be the integers that occur in (12); assume that
T>t.
Write also yasa canonic series
y = yo + yiP + yiP2 + --
and put, in analogy to x(n),
/n) = ^0 + yiP + • • • + yn- iPn~ 1 (n= 1, 2, 3,...).
Then, by the definition of T,
x(n) = y(n) f()r ,,= 1,2,...,7:

It is strictly differentiable 243
Apply the formula (9) twice for n = T, once with the variable x and
once with the variable y, and subtract the results. It follows that
FT(x)-FT(y) = (x-y)G(y^).
Here, by T > t and by (16),
\(FT(x)-FT(y))-(F(x)-F(y))\p<p-(T + v-s=\x-y\pp-s-\
and by (12),
\G(y^)-G(y)\p<p-s.
Therefore
\(F(x)~ F(y))-(x - y)G(y)\p<\x - y\pm<ix(p-s- l, p-%
whence
F(x) - F(y)
-G(y)
<P~5- (17)
P
x-y
In this inequality allow x to tend to the element y of /p, so that T
tends to infinity and thus is finally larger than any given integer t. It
follows that the integer s in (17) may be arbitrarily large, hence that
f F(x)-F(y)
hm = F (y)= G{y\ (18)
x -*■ y X y
which proves the following result.
Theorem 5 IfG is continuous on Ip, then its Dieudonne integral F
has a derivative Ff at all points of Ip, and F' = G everywhere.
As will now be shown, even more can be proved.
Denote by H : Ip x Ip -► Qp the function of (x, y)elpx Ip defined by
y(x)-F(y) for x^y,
H(x,y) = { x-y
G(y) for x = y.
This function H is continuous at all points (x, y) for which x ^ y; for
both its numerator F(x) — F(y) and its denominator x — y are
continuous on Ip x Ip, and x — y ^ 0. (We have proved such a result
for quotients of continuous functions only for functions in one
variable; see Chapter 8, Theorem 12. The assertion for H can easily be
proved directly.)
As we prove now, the function H is continuous also at all points

244 The Dieudonne integral
(Y, Y) where Ye/p, hence is continuous on the whole of Ip x Ip. For let
(x, y) be a variable point of Ip x /p which tends to the point (Y, Y) so
that both x and y tend to Y. lis and £ have the same meaning as in (12),
finally
\x-y\p<p'1 and \y- Y\p<p~\
If xi-y, then by (17),
|ff(x,);)-G0;)|p</?-s
and by the continuity of G,
\G(y)-G(Y)\p<p-\
so that
|ff(x, >;)-G(Y)|p </?"*. (19)
Owing to the continuity of the function G, the same inequality follows
also if x = y.
Since s may be arbitrarily large, by this inequality (19), H is then a
continuous function of (x, y)elp x Ip. This proves the following
theorem.
Theorem 6 If G is continuous on Ip, then its Dieudonne integral F
is strictly differentiate on Ip, with the derivative F' = G.
On restricting here the functions F and G to the set J, this result
implies that
if g is uniformly continuous on J, then its Dieudonne integral f is
strictly differentiable on J, with the derivative f' = g. (20)
8 A remark to Theorem 6
Let F and G be as in Theorem 6, and let C : Ip -> Qp be an arbitrary
pseudo-constant on Ip9
C(x) = 0 for xelp.
Also the function F + C is an integral of G because
(F + C)f = F' + C' = F' = G.
This new integral F + C need not be strictly differentiable on lp.
For let an, and cn, be the coefficients of F, and of C, respectively. As

Small values of the integral 245
just proved, the function F is strictly differentiable on Ip9 and therefore
(Chapter 13, Theorem 6),
lim n\an\p = 0.
n -*• oo
On the other hand, the pseudo-constant C can be chosen such that
limsup?2|cn|p= oo
«-*• 00
(Chapter 13, Theorem 7). Therefore also
limsupn|an + cn\p= oo,
«-*• 00
and since F + C has the coefficients an + cn, it follows that F + C is not
strictly differentiable.
Afar is it in general true that if F is strictly differentiable on Ip and if
F' = G, then F is the Dieudonne integral of G. For there are many
strictly differentiable pseudo-constants, and any of these may be
added to F without affecting its strict differentiability. On the other
hand, by its definition the Dieudonne integral of G is of course unique.
9 Every continuous function on lp has an arbitrarily small
integral
Let again F and G be as in Theorem 6. We found that
00
F(x) = ^ (x(k+1)-x(k))G(x(k)\ (15)
k = 1
and we now assert that here every partial sum
SN(x) = £ (x<* + x> - x«>)G(x«) (N = 1, 2, 3,...)
k= 1
is a strictly differentiable pseudo-constant.
For it is obvious from the definition that
SN(x) = SN(y) if \x-y\p<p-<N+1\ (21)
since by the upper bound for x — y,
x<*> = y*> for k = 1,2,..., N +1,
so that both sums consist of the same terms. This equation (21) shows
immediately that SN(x) is continuous and strictly differentiable on I
with the derivative S'N(x) identically 0.

246 The Dieudonne integral
Since G is assumed to be continuous on I it is bounded on this set,
say
\G(x)\p<p» for xelp.
Hence
00
X (x(fc + 1)-xw)G(xw)
k = N + 1
<p-N + u,
\F(x)-SN(x)\p =
uniformly in xelp. Here also
F{x)-SN(x) = <&(x\ say,
is an integral of G, and the integer N may be chosen arbitrarily large.
We thus arrive at the following theorem.
Theorem 1 Let Gbea continuous function on Ip, and let s be an
arbitrarily large positive integer. Then there exists an integral <X> of
G such that
|<D(x)|p</?-s for xelp.
An analogous result holds for functions g on J which are uniformly
continuous on J.
There is of course nothing similar in real analysis.
10 Every continuous function can be approximated by pseudo-
constants
Let G : /p -»Qp be a continuous function on /p, and let s be any
positive integer. There exists then a second positive integer t = t(s)
independent of x and y such that
\G(x)-G(y)\p<p~s if x,yelp and \x-y\p<p-\
As in Chapter 10 §5, we construct a step function S \Ip-+Qp which
approximates G, in the following manner.
Since xelp is a p-adic integer, there exists to it a unique rational
integer ye J satisfying the two conditions
\x-y\p<p~\ 0<y</?'-l.
With this choice of y put
S(x) = G(y).

The integral of the characteristic function 247
Since |x — y\ <p~\ it follows that for all xel ,
\G(x)-S(x)\p=\G(x)-G(y)\p<p-\ (22)
Since further numbers x, x0elp satisfying |x — x0|p < p~l correspond
to the same number y, the step function S has the property
S(x) = S(x0) if x, x0elp and |x — x0|p <p~*.
From this property it follows again that S is continuous and strictly
differentiable on I and is a pseudo-constant. In particular, we have
obtained the following result.
Theorem 8 Let G be a continuous function on /p, and let s be an
arbitrarily large positive integer. Then there exists a pseudo-
constant C(= S) on Ip such that
\G(x)-C(x)\p<p~s if xelp.
For the same reasons, functions g uniformly continuous on J can be
approximated arbitrarily closely by pseudo-constants on J.
11 The Dieudonne integral of the characteristic function of a
ball
Let
U(a; s): \x —a\p<p~s
be a ball in I where aelp, and s is some positive integer. (The case
when s = 0 is trivial since U(a; 0) is identical with Ip.) As in Chapter 7
§ 4, the p-adic integer a may be replaced by an integer A in J so that
U(a; s)= U(A; s). For this purpose take for A the unique rational
integer for which
\A-a\p<p~s, 0<A<ps-l.
Denote by X(x) the characteristic function of U(A; s); thus
(1 if xel„ and \x- A\„<p~s,
[0 otherwise.
It is again clear that X(x) is continuous and strictly differentiable and
is in fact a pseudo-constant.
Let now F : /„ -» QD be the Dieudonne integral of X(x). If, as before,
x = x0 + xyp + x2p2 + ...

248 The Dieudonn£ integral
is the canonic series of x, and if further
*(k) = x0 + x1p + .... + xk_1pk~1 (k = 1, 2, 3,...),
then
00
F(x)= X (x(k + 1)-x(k))X(x(k)).
k= 1
We shall now evaluate explicitly the sum on the right-hand side.
Two main cases will be distinguished where A = A0 + Alp + ...
+ As_lPs-\
Case 1 A = 0, hence A^ = Al = ... = As _ x = 0.
Then x lies in 1/(0; s) if also
Xq —— Xi —— . • . — X„ i — v.
If these equations hold, then also all the derived numbers x(k) lie in
1/(0; s). It follows therefore that
00
F(x)= ^ (*(fe + 1)-x(fe))x l = x-x(1) = x if xe[/(0;s). (23)
k= 1
Next assume that x does not lie in 1/(0; s). There is then a smallest
suffix r satisfying
0 < r < s - 1, xr ^ 0.
If r = 0, none of the numbers x(k) belongs to 1/(0; s), and therefore
F(x) = 0 if r = 0 and x^£/(0;s). (24)
If r > 1, then
but not
(1) (2) (r)
(r + 1) (r + 2) (s)
-A' ^ -A' ^ • • • 2 -A' J
belong to 1/(0; s). Hence
F(x) = £ (x(fe + X) - x(fe)) xl = x(r + 1)- x(1) = xr//
fe = l
if r>\ and x^£/(0;s). (25)
Case 2 ^4 7^ 0.
Denote by r the largest suffix for which
Hence
1 < r < s and A = A(r\

The special balls U{m;M) 249
First assume that x lies in U(A; s). This requires that
xk = Ak for k = 0, 1,..., s — 1,
hence that
Xr — X~ _j_ i — . . . — Xs i — VJ 11 I "\ S.
It follows that x(k) lies in U(A;s) for all k>r; moreover, x(r) = A.
Hence
00
F(x) = £ (x(k +1)- x(k)) xl=x- x(r) = x-A if xe U(A; s).
k = r
(26)
Secondly, assume that x does not lie in U(A; s), but that at least one
of the numbers x(k) does so. This can happen only if r < s and if there is
a suffix a satisfying
r < o <s
such that
Xr = Xr + i = . . . = Xa _ 2 = w, ^a-i 7^0.
Then
XW = x(r + l) = = x{a- l) = AeU(A; s), x(<t^U(A; s),
whence
<T- 1
F(x) = X (x(k +1)- x(k)) x 1 = x{a) - x{r) = x(a) -A
k = r
if x£U{A\s). ill)
The Dieudonne integral of the characteristic function X(x) of the
general ball U(A; s) has now been determined in all cases, and the
results are given by the formulae (23)-(27). It is particularly
interesting to find that this integral F(x) may in certain cases be
distinct from 0 even when x does not lie in U(A; s).
12 The special balls U(m; M)
For the next chapter we require the Dieudonne integrals of the
characteristic functions of a special sequence
{U(m;M)} (m-0, 1,2,...)
of balls in I Here the integer M is defined as follows.

250 The Dieudonne integral
Take
M=l if m = 0. (28)
If, however, m is a positive integer, then let M be the integer for which
pM-1<m<pM-l. (29)
We shall use the notations
X(x9 m) and F(x, m)
for the characteristic function and its Dieudonne integral of the ball
U(m; M).
In the case of the ball 1/(0; 1) the integer r vanishes, and F(x; 0) is
given by the formula (23). If, however, m > 1, then it follows from (29)
that the integer r belonging to U(m; M) is equal to M — 1, and hence
there is no a. Hence in either case F(x, m) vanishes when x does not lie
in L/(m; M), and so §11 establishes the following result.
Theorem 9 For every non-negative integer \m the Dieudonne
integral F(x, m) of the characteristic function X(x, m) of the ball
U(m\ M) has the value
F(x, m) = X(x, m)(x — m),
both when x does, or does not, lie in U(m; M).
It is obvious that the restriction/(x, m) of F(x, m) to J is determined
by the same formula.
13 Problems
1 Independently of the considerations in § 3, prove directly that
1 00
h(x) = - - £ x2nP2»
Zn = 0
is a pseudo-constant on J.
2 Letx = x0 +xxp + x2p2 + ... be the canonic series of xelp, and let
r > 0 be a fixed integer. The function G :Ip->Qp is defined by
G(x) = xr;
determine the Dieudonne integral of G.
3 Determine the most general function g : J -+Qp which is identical
with its Dieudonne integral.

Problems 251
4 For x e J put
0(0) = 0 and g(x)=l/\x\p if x^O.
Evaluate the Dieudonne integral of g.
5 Let g : J -^ Qp and gm\ J -^ Qp where m = 1,2, 3,... be functions on
J such that
Km 0m(x) = sr(x) for xeJ.
m -*• oo
Prove that the Dieudonne integrals f oi g and /m of #m satisfy
lim /m(x) = /(x) for xeJ.
m
m -*• oo
Under which additional conditions on the functions is the second
limit uniform in x?
6 {cn} is a sequence of p-adic numbers satisfying
\cn\p<p~n (w= 1,2,3,...).
For any continuous function G :Ip-+Qp define F* :/ ->Qp by
00
F*(x)= X (x"+1>-xW)(G(x«)+4
n= 1
Is F*(x) an integral of G(x) (i.e. is F(x) = G(x) for xe/J?

16
The van der Put series and integral
Up to now, in the study of functions of a p-adic variable, much use was
made of their representation as interpolation series.
There also exists a very different kind of development of such
functions due to van der Put (see van Rooij & Schikhof 1971) which
may usefully be applied instead and which will form the subject of this
chapter. It has in particular the great advantage of giving an explicit
formula for a vast class of pseudo-constants.
The notation in this chapter will be the same as that in the last one.
Thus, to begin with, g : J -^ Qp denotes an arbitrary function on J
which need not be continuous anywhere. Later, however, g will be
assumed to be uniformly continuous on J and to be in fact the
restriction
g(x) = G(x) if xeJ
of a continuous function G :/„-►£> on I.
P *^P P
1 Properties of the canonic series of xe J
Every integer x in J can be written as a canonic series
x = x0 + xxp + x2p2 +. ..
with coefficients xn that are digits 0, 1,. .., p — 1. Only finitely many
of these digits are distinct from 0, and on omitting all except possibly
the first vanishing terrrn the series for x takes the form
x = £ X(k)pn{k\ (1)
k= i
Here K is a positive integer which depends on x; x(l) may be any one
of the p digits 0,1,. .., p — 1, while x (2),..., x(K) are non-zero digits
1, 2,..., p — 1; and the exponents n(K) are non-negative integers
satisfying
0 = n(\) < n(2) < ... < n(K). (y\

Balls and characteristic functions 253
For x = 0, 1,..., p — 1 the development (1) reduces to its first term,
and K=\. For all larger values of x the integer K is at least 2. We
define a special function q : J -► Qp by
q(x) = x(K)pn(K).
It follows that always
q(x)<x, (3)
with equality only if either K ^= 1 and therefore x = 0,1,..., p — 1; or
if both K = 2 and x (1) = 0.
The partial sums in (1),
*D"]= t x(/c)p"w 0'= 1,2,...,K) (4)
fc= 1
will be called the segments of x. Evidently
q(x[JV = x(J)PnU) (J =1 2,..., K)
and therefore
x\J] - q(x\J~]) = x\J - 1] (j = 2, 3,..., K).
From (4),
0 < x[l] < x[2] < ... < x\_K\ (5)
and
q(x\j-])< x[j] < (P - 1) + (P - 1)P + (P - 1)P2 + • • • + (P - l)pn0)
= P»o-> + i_l (6)
Here the left-hand side may vanish if j = 1, and is otherwise at least
2 The balls u(m, M) and their characteristic functions x(x, m)
At the end of the last chapter we introduced the balls U(m, M) in I \
here M = 1 if m = 0, while for m > 1 the integer M was defined by the
inequality
pM~1 <m<pM-l. (7)
Let w(m, M) be the intersection of l/(ra, M) with J; it consists thus of
all x e J for which
\x — m\p<p~M.

254 The van der Put series and integral
The characteristic function #(x, m) of w(ra, M) is given by
fl if xej and \x — m\p<p~M,
[0 otherwise.
Thus x is the restriction of the characteristic function X of l/(ra, M)
to J.
Lemma 1 Every integer xeJ lies in exactly all those balls
u(m; M) for which m is one of the segments
x[fc] (£=1,2,. .., K)
of x.
Proof First consider w(0; 1). This ball consists of all non-negative
integral multiples of p. Hence x lies in w(0; 1) if and only if
x[l]=x(l)p° = 0 = m.
Secondly, let m > 1 so that M is connected with m by the
inequalities (7). The elements of u(m\ M) have the form
x = m + pMy, (8)
where y is a rational integer. This integer cannot be negative because
otherwise
x < m - pM < (pM - 1) - pM = - 1
by (7), contrary to xe J. Hence m is the smallest element of w(m; M),
and the elements of this ball form the arithmetic progression
m,m + pM, m + 2pM, m + 3pM,...
On comparing the form (8) of the elements of u(m; M) with the
expansion (1) of x and referring to the inequality (6), we see that x
belongs exactly then to u(m; M) when m is one of the segments x [fe].
Lemma 2. If y is a segment ofx e J, then all the segments of y are
also segments of x.
This is obvious from (1) and the analogous expression for y.
3 The van der Put series of g
Each characteristic function #(x, m) has the property
X(x, m) = x(y, m) if x,yeJ and \x-y\v<p~M.

The van der Put series of g 255
It follows that all these characteristic functions are uniformly
continuous and strictly differentiable on J and are in fact pseudo-
constants.
Let now g : J -► Qp be an arbitrary function on J. The van der Put
series of g is defined by the following theorem.
Theorem 1 There exists to g a unique sequence {B0, B^ B2,...}
of p-adic numbers such that
00
g(x) = ^ BmX(x>m) f°r xeJ- (9)
m = 0
Proof Since m is the smallest element of u(m; M) and therefore
x(x, ra) = 0 if m>x,
only finitely many terms in the series (9) are distinct from 0. There is
thus no problem of convergence.
The coefficients Bm can be obtained recursively by substituting
successively x = 0, 1, 2,... in (9). Since #(ra, m) = 1, each coefficients
Bm becomes then expressible in terms of the preceding ones. Indeed,
by Lemma 1,
9(x)= Z Bx[k] (x = 0,1,2,...). (10)
k= l
Here the suffixes of B run over all the segments of x.
These implicit formulae for the Bm can easily be replaced by explicit
ones. We know that K = 1 for x = 0, 1,..., p — 1; hence
Bm = g(m) for m = 0, 1, 2,..., p— 1. (11)
Next let K > 2. By Lemma 2, x[K — 1] has the segments
x[fc] (fc=l, 2,..., K-l),
and therefore, by (10),
g(xlK-l-])= X *,w.
& = l
On subtracting this equation from (10) it follows that
Bm = g(m)-g(m-q(m)) for m>p. (12)
The development (9) is called the van der Put series of g, and the
coefficients Bm are called the van der Put coefficients of this function.

256 The van der Put series and integral
By way of example, the function g(x) = x has the van der Put series
00
x= Z q(rn)x(x,m)
m = 0
and the van der Put coefficients Bm = q(m)9 as follows from (11) and
(12).
In a similar way as for the interpolation coefficients, we deduce
easily from (10), (11), and (12) that
max \Bk\p= max \g(k)\p (m = 0, 1, 2,. ..), (13)
0 < fc < m 0 < fc < m
and that therefore
wp\Bk\p=sup\g(k)\p. (14)
kej kej
Thus the coefficients Bk are bounded exactly when g is bounded on J.
4 The interpolation coefficients an (m) of t (jc, m)
It is of some interest to discover the relations between the van der Put
coefficients Bm of g and the coefficients bn in its interpolation series
9(x)= Z M ) for xeJ-
n=0 \nJ
For this purpose, let us determine the unknown coefficients an(m) in
the interpolation series
%(x, m)= Z an(m)\ ) for xeJ (^ = 0,1,2,..,).
n = 0 W
The easiest method for doing so is to use the generating series. Let
again z and Z be two indeterminates over Qp connected by the
equations
z = Z/(l+Z), Z = z/(l-z), (l + Z)(l-z)=l.
Then (Chapter 9 § 5), formally,
OO 00
Z x(rc,m)z" = (l + Z) Z an(m)Zn.
n=0 * n = 0
Here, on the left-hand side, %(n, m) vanishes unless n has the form
n = m + pMk, where keJ,

Relations between bn and Bm 257
and then % (n, m) = 1. Hence
00
X x(n, m)zn = zm(l + zpM + z2^ + z3pM + . . .),
n = 0
and it follows that
00
1 / Z \m f / Z VM / Z \2^M
/ z \3pM
By means of the binomial theorem, the right-hand side can be written
as a formal power series in powers of Z. A simple comparison of the
coefficients of the different powers Zn on both sides of this identity
leads then to the following results.
Firstly,
an(m) = 0 if 0<72<m-l. (15)
Secondly,
^™>-?(-ir"-^»+V*) ^ ^^ (16)
Here the summation extends over all rational integers k satisfying
0<k<p-M(n-m).
In particular,
am(m) =1 (m = 0, 1, 2,.. .).
By Chapter 10 §10, the coefficients an(m) allow the estimate
\an(m)\p<p-[in- ^/^" "I for all m, neJ. (17)
Naturally all these coefficients are rational integers.
5 Relations between the coefficients bn and Bm of g
The arbitrary function g : J -+Qp has the two developments
»(*)= t b«(j= t BmX(x,m)
n = 0 \n/ m = 0
into an interpolation series and a van der Put series when xeJ.
Replace here the characteristic functions by their interpolation series

258 The van der Put series and integral
It follows then that
K= t an(m)Bm (/1 = 0,1,2,...). (18)
m = 0
The formula (15) shows that the summation need only run from m = 0
to m = n.
We may consider (18) as a system of linear equations for the van der
Put coefficients Bm. Since an(n) = 1, it can be solved for the latter, in the
form
m
Bm= Z AJn)bH (ro = 0,l,2,...). (19)
n= 0
Here the new coefficients Am(n) are again rational integers, with
Am(m)= 1 (m = 0, 1, 2,.. .).
These Am(n) are in fact the van der Put coefficients of the
function I
\n
xx x
= E Am(n)x(x,m). (20)
m = 0
The formulae (11) and (12) allow us therefore to determine them,
Am(n)=[ J if m = 0,1,. ..,/7-1;
. , , /wi\ fm — q(m)\ .„
mW=L)~( n ) m~/7, (21)
By these formulae, always Am(n) = 0 if m < n.
6 Consequences of the continuity of g at a point v
So far, the function g : J -^ Qp has not been restricted in any way. In
much the same way as in the study of the interpolation series, let us
now impose suitable restrictions on g and determine their effect on the
van der Put coefficients Bm.
As a first hypothesis, assume that g is continuous at a certain point

Consequences of the continuity at y 259
y of J. This means that to every positive integer s there is a second
positive integer t = t(s) such that
\g(x)-g(y)\p<p~s if xeJ and \x-y\p<p-\ (22)
By the formulae (1) and (2) of §1, the canonic series for x can be
written as
x = £ x(k)pn«\
k=l
where the x(k) are digits 0, 1,..., p — 1 of which only x(l) may be
equal to 0, and where the exponents n(k) satisfy
0 = rc(l) < rc(2) < ... < n(K).
If y = 0, let already £ > 2, so that |x|p < p~ 2. It follows that y is the
segment
y = x[l]=0
of x and that therefore
n(k) >t (k = 2,3,...,K). (23)
If, however, y ^ 0, then let s and hence £ be already so large that
\y\P>p~t-
Since |x — y\p < p~ \ it follows that we are allowed to assume that y is
a segment
y = x[j~]
of x where j satisfies
2<j<K,
Now
n(k)>t (k = 7+ 1,..., K). (24)
We may consider (23) as the same set of formulae, but with j = 1.
On account of (23) or (24), we can now put x — y in the form
x-y= £ x(k)p"<*>
* = j + l
and more generally have
xiq-y= £ x(k)p"<*> (i =7 +1,..., K).
fc = j + l

260 The van der Put series and integral
Here, by (23) and (24),
\x\_i]-y\P<p-' (i = j + l,...,K),
whence, by (22),
\g(xli])-9iy)\P^P~s (i=j +1,..., K). (25)
Next, by (10),
g(x[}])= i Bxm (i =1, 2,..., K)
k= 1
and
j
g(y) = g(x\J~])= £ Bx[k}9
k= 1
so that
0(*H)-0()0= Z BxW (i=; +1,..., K). (26)
* = j + i
If X = 7 + 1, then by (25) and by x = x[X], it follows immediately
that
\BX\P<P~S. (27)
If, however, K>j + 1, the same result (27) is obtained by subtracting
the formula (26) for i = K—1 from the same formula for i = K and
applying (25) twice.
Conversely, if the inequality (27) is true for all suffixes x for which
|x — y\p < p~ \ then, on applying the equation (26) for i = K, the same
construction of/ leads to the result that \g(x) — #()01^ ^ P~s- Thus the
following property of g has been proved.
Theorem 2 The function g : J -^ Qp is continuous at the point
ye J if and only if
l^mlp->0 as m^=y and \m — y\p-^0. (28)
In Chapter 11, a similar Theorem 5 was proved for the
interpolation coefficients. However, this theorem was true only for
bounded functions.
It is clear from Theorem 2 that g is continuous at every point of J if
and only if (28) is satisfied for every ye J. This is thus the case if
g(x) = l/(x + 1).

The van der Put series on Ip 261
7 Consequences of the uniform continuity of g on J
The following theorem establishes the condition under which the
limit relation (28) holds uniformly in y.
Theorem 3 The function g : J ->QP is uniformly continuous on J
if and only if
lim Bm = 0. (29)
m -* oo
Proof If g is uniformly continuous on J, then to every positive
integer s there is a positive integer t = t(s) independent of x and y such
that
\g(x)-9(y)\P<P~s ^ ^y^J and \x-y\p^p-\ (30)
Now, by (12),
Bm = g(m)~ g(m —q(m)) for m>p.
Here
m — (m — q(m)) = q(m)
is a function of m which, by its definition in §1, tends to 0 as m tends to
oo. Hence an integer m0 exists such that
\q(m)\p<p~t if m>m0.
Therefore, by (30),
\Bm\p<p~s if m>m0,
proving the assertion (29).
Conversely, assume that (29) holds. Then the van der Put series
oo
m = 0
converges uniformly in x because
\%(x, m)\p < 1 for all x,meJ.
Since all the functions ^(x, m) are uniformly continuous on J, g(x) is
then likewise uniformly continuous on J.
8 The van der Put series on Ip
The characteristic function #(x, m) of the ball u(m, M) on J is the

262 The van der Put series and integral
restriction to J of the characteristic function X(x, m) of the ball
U(m, M) on I This suggests that we should consider the analogous
series
00
X BmX(x,m),
m = 0
where now x runs over the larger set Ip. Also such a series is called a
van der Put series.
We show by means of an example that such a van der Put series
may well converge everywhere on Ip even when its coefficients Bm do
not tend to 0.
Consider the series
00
G(x) = £ X(x, p")
n= 1
with the coefficients
fl if m — pn, 72=1,2,3,...,
m (0 otherwise.
The integer M belonging to m = pn satisfies
pM ~ 1 < pn < pM - 1
and hence has the value
M = n + 1.
Hence C/(m, M) is the ball U(pn;pn+1) and so consists of all xelp of
the form
00
k = n+ 1
where the xfc are arbitrary digits 0, 1,..., p — 1. From this it follows
immediately that x cannot lie in two distinct balls U(pn; pn+ x).
Therefore
Jl if x lies in some ball U(pn; pn+l\
[0 otherwise.
In particular,
G(0) = 0.

The series of a continuous function on Ip 263
On the other hand, pne U(pn; pn + *), and therefore
lim G(pn) = 1, lim pn = 0,
n -* oo n -* oo
showing that G is discontinuous at x = 0.
At every other point of I the function G is continuous. For if y ^ 0
lies in some ball U(pn; pn+ *), and x tends to j;, then also x finally
belongs to U(pn; pn+l\ and hence G(x) = 1 = G(y). If, however, y ^ 0
does not lie in any set U(pn; pn+ *), then the lowest term ynpn in its
canonic series has a digit factor yn equal to one of the digits 2, 3,...,
/7—1, and as x tends to y, finally the canonic series for x begins with
the same lowest term ynpn. It follows then that G(x) = 0 = G(y).
In the special case when /7 = 2, G(x) vanishes for x = 0 and is
otherwise equal to 1; hence x = 0 is still the only point of
discontinuity.
9 The van der Put series of a continuous function G on Ip
From Theorem 3 the following consequence may be drawn.
Theorem 4 Let G:Ip-^Qp be continuous on Ip, and let
00
G(x)= ^ BmX(x,m)
m = 0
be its van der Put series on Ip. Then
lim Bm = 0. (29)
m -*■ oo
Conversely, if (29) is true, then the series G(x) converges uniformly
on Ip and its sum is continuous on Ip.
Proof If G is continuous on I then its restriction g to J is uniformly
continuous on J, and (29) holds by Theorem 3. Conversely, if (29) is
satisfied, then the series G(x) converges uniformly on I because
\X(x, m)\p < 1 for xelp and me J.
Since the characteristic functions X(x, m) are continuous on I the
same is then true of the sum G(x) of the van der Put series.

264 The van der Put series and integral
10 The van der Put integral
Consider first an arbitrary function g : J -+ Qp on J, with the van der
Put series
00
0(a) = Z BmX(x,m).
m = 0
Definition 1 The new series
00
/(*)= I Bmx(x, m)(x - m) (31)
m = 0
is called the van der Put integral of g.
There is no problem of convergence because ^(x, m) = 0 for m > x.
Secondly, let G:I ->Qp be a continuous function on I and let
00
G(x)= £ BmZ(x,m)
m = 0
be its van der Put series.
Definition 2 Tte new series
00
F(x)= £ BmX(x,m)(x~m) (32)
m = 0
is ca/fed the van der Put integral of G.
Since G is continuous on I the coefficients Bm have the p-adic limit 0.
On the other hand,
\X(x, m)(x-m)\p < 1 for me J and xe/p,
and the functions X{x, m)(x — m) are continuous on Ip. Hence the
series (32) converges uniformly on /, and its sum F is a continuous
function on I
It is clear that if f and # denote the restrictions of F and G to J,
respectively, then f is the van der Put integral of g.
In the two definitions, we have still to justify the names of integral.
We shall prove the following two assertions.
Theorem 5 Let g : J -+'Qpbe any function on J, and let f be its

The van der Put integral 265
van der Put integral If g is continuous at the point y of J, then
f'{y) exists and is equal to g(y).
Theorem 6 Let G :Ip-^Qpbea continuous function on Ip, and
let F be its van der Put integral. At every point y ofIp the derivative
F'(y) exists and is equal to G(y).
It will be sufficient to prove Theorem 5 because the same method
leads to Theorem 6.
Proof of Theorem 5. Let s be an arbitrarily large positive integer. By
Theorem 2, there exists a second positive integer t = t(s) such that
\Bm\p<p~s if OKlm-y^Kp-* (33)
From the series for f and g,
fix) - /GO
x-y
00
-g(y)
V d (X(x,m)(x-m)-x(y,m)(y-m)
= L Bm\ x(y,m)
00 j£ — YYl
= £ Bm(x(x, m) - x(y> ™))
m = 0 x y
In the right-hand series all terms with %(x, m) = x(y, m)' vanish, and
hence only terms with ^(x, m) ^ %(y, m) need be considered.
Let us assume that already
^<\x-y\p<p~l.
If x(x, m) = 1 and x(y, wi) = 0, then
\x-m\p<p~M, \y-m\p>p~M,
hence
p~ M < \x - y\p = \y - m\p < p~ \
and therefore
M> t.
It follows that
x — m
x-y
< 1 and \y — m\D<p \

266
The van der Put series and integral
whence
Bm(x(x, m) - x(y, m))
x — m
x-y
<\BmL<P s.
If, however, #(x, m) = 0 and x(y, m) = 1, then
x-m.> p
M
\y-m\<p
M
hence
p M<|x->;| =|x-mL</? \
so that again M > t. Now
x — m
x-y
which proves (34) also in this case.
We have thus established that
■/(*) - /GO
= 1 and \y-m\<p \
-g(y)
<P
(34)
x-y
as soon as |x — y\ is sufficiently small. Since s may be chosen
arbitrarily large, this proves the assertion.
By making use of Theorem 4, Theorem 6 is proved in the same
manner.
11 The relation between the Dieudonne integral and the van
der Put integral
Two different definitions of an integral have now been given, one due
to Dieudonne and one to van der Put. Let them be distinguished by
the suffixes D and P, respectively.
By way of example, we found earlier that the function g(x) = x has
the two integrals
x
2 i oo
Z Zn = 0
where the xn are the digits in the canonic series for x, and
00
/p(*)= Z q{m)x(x,m\
m = 0
Generally, a function g :J-^QP has the two integrals
/d and/p

Dieudonnk integral and van der Put integral 267
which are well defined on J without any restriction on g; and a
function G :Ip-^Qp continuous on Ip has the integrals
FD and FP.
In the second case,
F'D(x) = F'p(x) = G(x) for all xelp9
and hence the difference
C(x) = FD(x) - FP(x)
is a pseudo-constant on I In the first case, the equation
f'D(x) = f'p(x) = g(x)
has been established only for those points x on J at which g is
continuous. Thus, although the difference
c(x) = fD(x) - /P(x)
is well defined on the whole of J, we cannot be certain whether the
equation
c'(x) = 0
is true everywhere on J, and so c may perhaps not be a pseudo-
constant on J.
In either case, the problem arises of determining the functions C
and c. The example above for g(x) = x suggests that this problem may
be difficult.
It may then come as a surprise that there is in fact a very simple
answer, as follows.
Theorem 7 If g : J -> Qp is an arbitrary function on J, then
/dM = fp(x) for all x E J-
Theorem 8 If G : /p -> Qp is a continuous function on Ip, then
FD(x) = FP(x) for all xelp.
Proof We first note that the four integrals depend linearly on g and
on G, respectively. Thus if gp (p = 1, 2,..., r) are finitely many
functions on J with the integrals/pD and/pP, and if yp (p = 1, 2,..., r)
are an equal number of p-adic constants, then YJ0 = \yPQP nas tne two

268 The van der Put series and integral
integrals Ep = l ypfpD and E^ = 1 ypfpF, This property is obvious from
the definitions of the Dieudonne and van der Put integrals. An
analogous rule holds for the integrals of Ep = l yp Gp. It is further clear
that the two integrals of the function 0 are themselves equal to 0.
First consider the integrals fD and fP of an arbitrary function
g :J -^>Qp. By Theorem 1, g can be written as a van der Put series
00 X
9(x)= E 5m/(x,m)= E BmX(x,m) for xeJ;
m = 0 m = 0
the second representation holds because ^(x, ra) = 0 if m > x. By
Definition 1, g has the van der Put integral
X
/M = E BmX(*, rn)(x -m).
F m- 0
Next, by Theorem 9 of Chapter 15, the characteristic function
X(x, m) of the ball (7(m, M) has the Dieudonne integral
X(x, m)(x — m).
On taking the restriction to J, it follows that the characteristic
function #(x, m) of w(m, M) has the Dieudonne integral
X(x, m)(x~m).
Hence, on applying the van der Put development of g, we see that g
has the Dieudonne integral
X
E BmX(x9m)(x-m)9
m = 0
which is equal to the van der Put integral. Thus fD(x) = fF(x) for all
xgJ.
In order to prove the analogous result for continuous functions G
on Ip, denote by #,/D, and/P the restrictions of G, FD, and FP to J,
respectively. Here G, FD, and FP are continuous on Ip, and therefore g,
/D, and /P are uniformly continuous on J. By what has just been
proved,
/dW = /p(x) for *eJ-
This, however, implies that also
FD(x) = FP(x) for xe/p
since, e.g. all four functions/D,/P, FD, and FP have identical van der
Put development.

An arithmetic property of the integral 269
Theorem 7, combined with the Theorems 1 and 2 of Chapter 15,
shows also that the van der Put integral /P is continuous at y e J if g is
bounded at y, and that/P is uniformly continuous on J ifg is bounded
on J.
12 An arithmetic property of the Dieudonn£-van der Put
integral
The arbitrary function g : J -+Qp has a unique Dieudonne-van der
Put integral fD =/P, or say /DP. More general integrals will be
obtained by adding arbitrary pseudo-constants. As we shall now
prove, the special integral /Dp has an interesting arithmetic property
which at least partially distinguishes it from other integrals.
Let, as in § 5,
x /x\ x
9(x)= Z K( = Z BmX(x>m)
n=0 \n/ m = 0
be the interpolation series and the van der Put series of g on J. We
found the two formulae
m
K= Z a„(m)Bm and Bm= £ Am(n)b„
m = 0 n = 0
connecting the two types of coefficients; here an(m) and Am(n) were
rational integers which had the explicit values
fl» = E(-ir"W "m,^ if m<n,
m + pM/c
and
Am(n)==[n) if ^ = 0, 1,..., p-1;
/m\ (m-q{m)\
respectively.
Consider next the interpolation series,
X (x, m) (x - m) = ^ a. (m) (* ) say,
for the van der Put integral of x(x, m). A proof exactly like that in § 4

270 The van der Put series and integral
leads to the explicit formula
a,.(,n) = p*p(-ir—- (jpMk),
where, just as in the equation for an(m\ the summation extends over
all suffixes k satisfying
0<k<p~M(j-m).
Now denote by cn the interpolation coefficients of the integral /Dp,
so that
00 00 I y
/dpW= Z Bmx(x, m)(x - m) = Z cn
m=0 n=0 \n
On substituting the series for #(x, m)(x — m), we find that
oo oo /v\ oo oo m ,^.
/dpW= Z Z B„«i(«) . = Z Z Z Am(n)b„aj(m)( .
m = Oj = 0 V / « = 0j=0b=0 \J
whence
oo m
cj= Z Z Am(n)&j(m)bn.
m = 0 n = 0
This formula leads to the following conclusion.
Theorem 9 If g '• J ^ Qp has the interpolation coefficients bn, and
its integral fDP has the interpolation coefficients cn, then the cn can
be written as linear forms in the bn with rational integral
coefficients.
There is no analogue of this theorem in real analysis where we have
integral formulae like
X \ . / X \ 1 X
l/dX=2 +2 21+COnStant'
and
with coefficients that are fractional rational numbers.
I do not know whether the property in Theorem 9 characterises the
integral /Dp in the set of all possible integrals of g.

A formula for pseudo-constants
271
13 A formula for pseudo-constants
Consider again an arbitrary function g : J -+Qp and its van der Put
series
00
9(x)= X Bmx(x,m).
(9)
m = 0
For x, ye J and x ^ y,
9<x)-g(y)
00
x-y
= 1B
m = 0
x-y
and therefore
9(x)-g(y)
x-y
< sup
p me J
B
X(x, m) - x(y, m)
m
x-y
Under the supremum sign only terms with #(x, m) ^ %(y, m) can
make a positive contribution. Therefore, without loss of generality, it
suffices to consider those terms in (9) for which
X(x, m) = 1, %(y, m) = 0,
since we may otherwise interchange x and y.
Denote again by T the integer for which
(35)
\x-y\» = p T-
By (35),
hence
whence
Since also
this implies that
x — m\ <p M and \y — m\>p
M
-M
<\y-m\=\x-y\' = p i,
M>T+1.
,M- 1
< m < pM - 1
m>pT.
Terms that belong to suffixes < pT make no positive contribution to
the supremum.

272 The van der Put series and integral
Since then
= \Bm\pxpT<m\Bm\p,
p
it follows that
->0 as x, ye J and 0 < |x - y\p^>0, (36)
p
provided that
m\Bm\p^0 as m->oo. (37)
This property (37) implies in particular that the series (9) for g(x)
converges uniformly on J, hence that g is uniformly continuous on J.
Therefore g can be extended to a continuous function G :Ip~+Qp with
the van der Put series
oo
G(x)= X BmX(x,m) for xel„.
m = 0
The proof of (36) may be repeated, but with G and X instead of g and
X, and leads to the result that
->0 as x,yElp and 0 < |x — y|p—>0, (38)
p
provided again that Bm satisfies (37). Here the quotient
G(x) - G(y)
x-y
is a continuous function of (xi, y)elp x I as long as x ^ y, and it tends
to 0 uniformly in (x, y) as 0<|x —j;| ->0. Hence the following
theorem has been proved.
Theorem 10 Assume that
p
lim m\Bm\p = 0.
m -*■ oo
Then the function g : J -+Qp defined by
00
9(x)= ^ BmX(x,m)
m = 0
is uniformly continuous, strictly differentiable, and a pseudo-
B
X(x, m) - x{y, m)
x-y
x-y
G(x) - Gjy)
x-y

jrroDierns
Z/J
constant on J, and the function G :Ip^>Qp defined by
00
G(x)= £ BmX(x,m)
m = 0
is continuous, strictly differentiable, and a pseudo-constant on Ip.
This theorem gives a sufficient condition for g and G to be pseudo-
constants. A necessary condition involving the coefficients Bm does
not seem to be known, and I do not know of any necessary or
sufficient condition in terms of the interpolation coefficients.
14 Problems
1 Let G : Ip -> Qp be the function
fO if x = 0,
Determine the van der Put series and the van der Put integral of G.
2 Determine the van der Put series of the function q: J -+Qp as
defined in §1.
3 Assume the function G :Ip-+Qp is continuous on Ip and has here
the van der Put series
00
G(x)= £ BmX(x,m).
m = 0
Discuss the continuity &nd repeated differentiability of the function
H:Ip^Qp defined by
00
H(x)= £ BmX(x,m)(x-mr.
m = 0
4 For xgJ and a given positive integer N let d(x) be the rational
integer satisfying
0 < d(x) <pN-l, \x- d(x)\p <p~N.
Determine the van der Put series of the function d : J -> Qp.
5 Let xeJ have the canonic series x = x0 + xyp + x2p2 + • • • • We
proved that the Dieudonne integral and hence also the van der Put
integral of g(x) = x is the function
1 00
f(x) = ±x2 + h(x), where h(x)= -- £ x„V",

274 The van der Put series and integral
and that x has the van der Put series
00
x= Z q(m)x(x>ml
m = 0
Use these facts to sum the two series
00 00
£ mq(m)x(x,m) and £ q(m)2x(x,m).
m = 0 m = 0
6 Let G : /D-> QD be the function
00
G(x)= £ p"X(x,p").
n = 0
Decide whether the derivative G'(x) exists, (i) when x ^ 0, and (ii)
when x = 0.

17
Definite integrals and difference equations
1 Definition of the definite integral
The two integrals by Dieudonne and van der Put considered in
Chapters 15 and 16 correspond to the indefinite integral (primitive or
anti-derivative) of real analysis. We proceed now to the study of a
p-adic analogue of the definite (Riemann) integral of real analysis.
If G : i -+R is 3. real continuous function on the closed interval
i: 0 < x < 1,
its definite integral over i is equal to the real limit
C fl 1 n- 1
G(x)dx =
\v
G(x)dx = lira - £ G(k/n)
0 «-*• oo TZfc = 0
where the points k/n divide i into n equal parts.
Let now G : In -+ Qn be a continuous function on I. This function is
determined completely if we know its values on the subset J ofIp. In
analogy to the definition in the real case, it makes therefore good
sense to define as the p-adic integral of G over I the p-adic limit
G(x)dx= lim - £ G(k) (1)
o<Mp->o nk = o
provided that this limit exists. Here the positive integer n tends to
infinity in such a manner that the highest power of p dividing n also
tends to infinity. The meaningless factor dx has been added under the
integral sign in order to display the variable x over which the
integration is carried out. Thus, if y and co are two p-adic constants
such that
xoo + yelp if xe/p,
then, by (1),
p in_1
G(xcd + y)dx = lim - X G(kco + y\ (2)
, o<\n\p->o nk = 0
~ p
again provided that the limit exists.

276 Definite integrals and difference equations
As we shall see, the problem of the existence and the properties of
the p-adic integral are closely connected with the behaviour of the
solutions F(x) of the difference equation
F(x + co) - F(x)
00
= G(x). (3)
The easiest method for studying these solutions is by using a slightly
generalised form of the interpolation series.
2 The generalised interpolation series
Let again G : Ip -> Qp be a continuous function on Ip. Further let co,
called the parameter, be a p-adic number satisfying
0<\oo\p< 1.
For every p-adic number a ^ 0 denote by odp the set of all p-adic
numbers x such that |x|p < |a|p. Thus oolp is a subset of Ip, and Ip is a
subset of co" 1lp. In particular, ooIp is identical with Ip exactly when
\co\p=l.
By its definition, the function G(cox) is defined and continuous on
co ~ lI and hence also on I hence can be written as an interpolation
p ,.. p
series
00
G(atx)= ^ &„,»(*) for X€/
i P'
n=0 v«'
Here the coefficients few n, which depend on both co and n, are given by
K,n= £ (- ir-*HG(a)*:) (n = 0, 1, 2,...).
By its continuity, G(cox) has the property (N),
lim fe« M = 0.
co, n
n -*• oo
Replace finally cox again by x in the interpolation series for G(cox).
We obtain then for G(x) itself the required generalised interpolation
series
G(x)= t b(X,0i] for xecoIp (4)
n=0 \ H J
which depends on the parameter co. When |co| < 1, ooIp is a proper
subset of /, and it may not be true that (4) is valid for all x in I .

Solutions of the difference equation 277
3 The continuous solutions of the difference equation
Let G\Ip^y Qp be the function just considered, and let F : cc>Ip -> Qp be
a continuous function on a>Ip which satisfies the functional equation
(3). The derived function F(cox) is continuous on I hence can be
written as a convergent interpolation series
F(cdx)= £ aw>n( ] for xe/
with the coefficients
p
n=0 v"'
'n
flM= I (-ir-fe( jF(cofc) (.2 = 0,1,2,...),
fc = 0
which satisfy the limit relation
p
lim £* „ = 0.
co, n
n -*• oo
On changing back to the variable x,
00 (x/co\
F(x)= £ awJ for xeco/^
n=0 \ H J
The difference equation (3) states that
F(x + co) — F(x) = coG(x).
Now, in the series for F(x),
\x + co)/co\ _ fx/co _
, o )-[o
and
(x + co)/co\ fx/co\ ( x/co
n ) \ n J \n— 1
Hence
for n > 1.
00 / x/cO \
F(x + co)-F(x) = £ flffl,»( _j 1 for xeo)/,.
On comparing this development with the analogous series for coG(x),
it follows that
aco,n+l=™bo,n (n = 0, 1,2,...),
while the first coefficient aa) 0 remains undetermined. Thus the
following result is obtained.

278 Definite integrals and difference equations
Theorem 1 The most general solution F(x) of the difference
equation (3) which is continuous for xecoIp has the form
F(x) = F(x\co) + a(Ot0,
where
00 I x/cO \
F(X\co) = coZK,n[n+l} (5)
while aw 0 is an arbitrary function of co.
We call F(x\(jo>) the basic solution of (3). It is characterised by the
equation
^(01^) = 0, (6)
as follows immediately from (5).
4 The existence and the value of the definite integral
As will now be proved, the existence of the definite integral depends
on the differentiability of the solutions of the difference equation (3).
Theorem 2 Let G : Ip -> Qp be a continuous function on Ip, and let
cd and y be two p-adic numbers satisfying 0 < \co\p < 1, yecoIp.
Let further F(x) be any solution of the difference equation
F(x + co) - F{x)
CO
which is continuous on coIp.
Then the integral
= G{x)
h
G(cox + y)dx
exists and is equal to F'(y) if and only if this derivative exists.
Proof Evidently
-"X G(tok + y)=~YJ (F((k+l)co + y)-F(kco + y)
n k = o ojn kf0 \
_F(con-\-y)-F(y)
con

Existence and value of the definite integral 279
As 0 < \n\ -> 0, the left-hand and right-hand sides tend to J7 and F\y\
respectively, if one and hence both limits exist. (Just as in the proof of
Lemma 1, p. 190, the existence of F\y) in J implies that in Ip, and vice
versa). The derivative evidently does not depend on the special choice
of the solution F(x).
Corollary Denote by F(x) an arbitrary solution of
F(x+l)-F(x) = G(x)
which is continuous on Ip. The integral
G(x)dx
i
p
exists and is equal to F'(0) if and only if this derivative exists.
In the theorem, the solution F(x) may be identified with the basic
solution F(x\co), in particular with F(x\ 1) in the corollary. Theorem 1
of Chapter 13 allows us to state a necessary and sufficient condition
for the existence of the derivatives F'(y\co) and F'(0|1). In analogy to
the notation an(y) of Chapter 13 put
00
bco,n(y)= L bco,k + ni fc ) (^ = 0,1,2,...).
Then F'(y\co) exists and has the value
oo / iyi
F'(y\co) = co X ±—LbatH{y)
^ — n n + 1
n=0 ,l +
if and only if
p b (v)
lim^^ = 0.
n -*• oo n ~r 1
In the special case when y = 0 this condition simplifies to
p b
lim -^- = 0. (7)
n -*• oo ^ ~r ^
The very similar equation
P b
lim -^- = 0 (8)
n -*• oo n
was the necessary and sufficient condition for the existence of the

280 Definite integrals and difference equations
other derivative G'(0). We note that neither of the two conditions (7)
and (8) implies the other one. This is clear for cd = 1 by the two
examples
^)=1/(/) hence F(*|l) = £•£
and
G(x)= IP-Li ^nce F(x|l) = I W
r = 0 \P J r = 0 \F+1,
For in the first one G'(0), but not F'(0|1), and in the second one
F'(0|1), but not G'(0), exist, as follows from Chapter 13 § 8.
Thus the differentiability of G(x) does not necessarily imply that of
F(x\cd), nor thus the existence of the integral. But it is even worse:
F'(0|1) need not exist even if G has continuous derivatives of all
orders. This follows from the Weisman example of Chapter 13 §13, if
the sequence {u-} in this example increases sufficiently rapidly.
However, the following result holds.
Theorem 3 Assume that the function G(x) is strictly differenti-
able. Then, if cd and y are as in Theorem 2, the integral
G(cox + y)dx
exists.
h
Proof. By the hypothesis, there exists a function H :Ipx Ip^Qp of
two variables continuous on I x I such that
= H(x,y) for (x,y)eIBxIB and x^y.
x-y
If 0< \cd\p< 1, then also
G(cox) — G(coy)
= cdH (cox, coy) for (x,y)eIDxID and x^y.
x-y
Here cdH(cdx, coy) is a continuous function on I x I and therefore
G(cdx) likewise is strictly differentiable and so in particular
continuous on I .
It follows then from Theorem 6 of Chapter 13 that
lim n\K,n\P = Q-
n -*■ oo

Identities for F(x\co) 281
Theorem 1 and the series (5) for F(x\co) lead therefore to the following
result.
Corollary If the function G(x) is strictly differentiable on Ip, then
so is every solution F(x) of the corresponding difference equation
(3).
Having thus established the close connection between the definite
integral and the difference equation, we proceed now to a deeper
study of the latter.
5 Identities for F(x | m)
The next considerations are based on the book by Norlund (1924)
where the difference equation is studied in the real and complex cases.
Denote by AT a positive integer which is not divisible by p, a
restriction which ensures that x + (k(D/N)ea>Ip if xea>Ip and keJ.
Put now
N - 1 /
F1(x)= £ F[x +
k = 0 \
kco
60
N
co) and F2(x) = F\x
By the difference equation (3) and a simple calculation,
FJ X+—: J -^(^) = ^(^ + 60160)-^(^60):= coG(x)
and / 6o\ 60
It follows that the function
F1(x)-NF2(x) = F*(x), say,
has the periodicity property
F*(x + (6o/A0) = F*(x) for xgcdIp.
It assumes therefore one and the same value, AN(co) say, at all the
points
nco/N (n = 0, 1, 2,...)
which, by pfN, lie dense on a>Ip. Since F* is continuous on a>Ip, it
follows that
F*(x) = AN(co) identically in xea>Ip.

282 Definite integrals and difference equations
This is equivalent to the first identity
> F\ xH
^ I N
co\ = NF[x
k = 0
On putting x = 0, it follows that
7cco
]+AN(co) for xEcoIp if piN. (9)
"-1 Aco
CO
Next replace in (3) the parameter co by — co. The difference equation
becomes then
F(x — co | — co) — F(x| — co) = — coG(x),
which is the same as
F((x — co) + co| — co) — F(x — co| — co) = coG(x).
This means that both F(x|co) and F(x — co\ — co) satisfy the difference
equation (3), hence, by Theorem 1, differ only by a quantity £(co)
independent of x. We thus obtain the second identity
F(x-co\-co) = F(x\co)-\-B(co) for xecoIp. (10)
On putting here again x = 0, it follows that
B(co) = - F(co\co) - - coG(O).
The first identity was proved only under the restriction that p/JV.
This restriction can be omitted when F(x\co) exists for all x in Q say
is a polynomial in x.
In order to distinguish it from the basic solution F(x\co) of (3), let us
denote by F(x\ co) a general solution of (3) which is continuous for all
xecoIp but perhaps is not continuous as a function of co. By Theorem
1,
F(x',co) = F(x\co) + aa>t0,
where aw 0 is an arbitrary function of co. It follows immediately from
(9) and (10) that also F(x; co) satisfies a pair of identities like (9) and
(10), but with different accessory terms ^4N(co) and £(00).
6 The existence of normed solutions
A solution F(x; co) of (3) is said to be normed if the accessory terms in
its two identities vanish for all allowed values of co. It will now be
proved that there always exist such normed solutions.

Existence ofnormed solutions 283
We first note that there always exists to the given function
G.Ip^Qp continuous on Ip a second function *F : Ip -> Qp which is
continuous and strictly differentiable on Ip and has the property that
its derivative
¥'(*) = G(x) for xelp. (11)
Simply take for *F the Dieudonne-van der Put integral of G. For then
(11) is satisfied by Theorem 6 of Chapter 15.
The assertion on normed solutions is now contained in the
following theorem.
Theorem 4 Let G :Ip-^Qp and *¥ :Ip->Qp be two continuous
functions on Ip such that *F is strictly differentiable on Ip and
satisfies (11). Let O(x|co), forO < \oo\p< 1, be the basic solution of
the difference equation
<S(* + a,)-0(x)
00
Then the derivative <J>'(x\co) exists and is a normed solution of
F(x + oo)-F(x)
00
= G(x).
Proof On applying the considerations in the proof of Theorem 3, it
follows, firstly, that also ^(cox) is continuous and strictly
differentiable on Ip. It therefore can be written as a convergent interpolation
series
'X
¥(a>x)= £ cmtH[ ) for xe/j
with the coefficients
n = o \ w
n
c», - = E (- !)""k[kJ^M) (« = 0, 1,...),
which, by the strict differentiability of *F(cox) and by Theorem 6 of
Chapter 13, satisfy the limit relation
lim"|c<o,»lP = 0- (13)
n -*• oo
On replacing cox again by x, it follows next that
00
*(*>= ic»..(T) for xewI°-
n = 0 \ n

284
Definite integrals and difference equations
Hence, in analogy to the considerations in §3, the basic solution of
(12) can be written as the interpolation series
£ ( X/™ \ c
O(x|co) = 60 X Cco,n[ .J f°r XECDlp.
n = o \n-\-1/
By (13), also this solution is strictly differentiable on ooIp. As the basic
solution of the difference equation (12), it satisfies by § 5 the two
identities
N-l
N - 1 /
I * x +
k = 0 \
koo
N
oo = NO x
00
N
+ X*(a)) for xecolp if p/JV,
(14)
and
0(^-601-60) = 0(^160) + ^(60) for xe^/p, (15)
where the accessory terms are given by
/C60
N- 1 /
x*(o))= X a> -
fe = 0 \
AT
60
and 5*(60) = - 0(60160) = - o)*F(0).
On differentiating the difference equation (12) with respect to x, it
follows that the derivative
O'(x|6o), =F°(x|6o) say,
is a solution of the original difference equation (3). Moreover, the
differentiated accessory terms in (14) and (15) vanish, and it follows
that F° satisfies the identities
koo
N- 1 /
Z F°(x +
fe = 0 \
N
aj\ = NF°[x
oo
N
for xeaolj, if p/JV, (16)
and
(17)
F°(x-6o|-6o) = F°(x|6o) for xeooIp.
This concludes the proof.
Theorem 4 suggests that it may be advantageous to study not a
single difference equation (3), but an infinite system of such difference
equations
Fs(x + 6o)-Fs(x)
60
= GAx)
(s = 0, 1, 2,...)
such that any two consecutive right-hand side functions
Gs(x) and Gs + 1(x)

Bernoulli polynomials and numbers 285
are related by identities of the form
GM(x) = yMG'M+1(x) or Gs+l(x) = SsG's(x).
Here the coefficients ys and Ss are independent of x, but may possibly
depend on both co and s. We shall in the remainder of this chapter
study two such sequences of equations.
7 The Bernoulli polynomials and the Bernoulli numbers
The first sequence is defined by
Gs(x) = sxs~1 (s=l,2,3,...)
so that
G; + 1(x) = (s + l)G,(x). (18)
With each Gs we associate the difference equation
D,:F-(* + °"-F-W=0,(»).
CD
where now both x and co ^ 0 may be anywhere in Qp and are not
restricted to Ip because we are dealing with polynomials. Being of
degree s — 1, the polynomial Gs can be written as the finite
interpolation series
n=0 \ n' y
with the coefficients
b%n = so°-1 £ (- 1)-~ k [Hk) ks ~ ' (n = 0, 1,..., 5-1)
k = 0
.s- 1
which are rational integral multiples of co'
By the construction in §3, D has the basic solution
^1-)-1^.(.^,
(s = 1, 2, 3,.. .)
which is a polynomial of degree s. In terms of this function, put now
fls(x|©) = (s+1)-^^1(0) (5 = 0,1,2, ) (19)
where the differentiation is as usual with respect to x; then Bs is a
polynomial in x of degree s. The difference equation Ds +1forFs+1

286 Definite integrals and difference equations
implies by (18) that
F's+i(x + co\co)-F's + 1(x\co)
CO
so that, by (19),
Bs(x + co\co) — Bs(x\co)
= g;+1(x) = (s + i)g,(x)
CO
= Gs(x) (5 = 0,1,2,...) (20)
if we put G0(x) = 0.
It follows now from Theorem 4 that Bs(x\co) is a normed solution of
Ds, hence that
N-l /
I B.(x +
k = 0 \
kco
N
co I = NBS[ x
CO .
f and ^(^-0)1-60) = ^(^160)
(5 = 0,1,2,...). (21)
Here, by the remark in §5, N may be any positive integer since Bs is a
polynomial.
We finally put
Bs(x) = Bs(x\l) and Bs(0) = Bs
(s = 0, 1, 2,.. .) (22)
so that, by the construction, Bs(x) is a polynomial in x of degree s with
rational coefficients, and Bs is a rational number. In the usual
terminology, Bs(x) is the Bernoulli polynomial and Bs the Bernoulli
number of suffix s.
From the interpolation series for Fs + 1 and the explicit formula for
its coefficients, it is clear that
Fs+ i(*M = cos+1Fs+1(x/co\l),
whence, on differentiating with respect to x, by (19),
£s(x|6o) = 60s£s(x/6o) (s = 0, 1, 2,.. .), (23)
so that the identities (21) may also be written as
Bs(Nx) = Ns-lN^Bs(x + ^) and Bs(l - x) = (- l)s£s(x).
(24)
For the lowest suffixes,
£0(x)=l, B1(x) = x--|, ,B2(x) = x2-x + i
B3(x) = x(x-l)(x-\),

The integral of a polynomial
287
and
^o= 1> Bi= ~ 2> ^2 = 6» ^3 = ¾ .84=-30, #5 —0,
B6 = 42> Bl = 0,
^8 =
30"
8 The integral of a polynomial
The difference equation Ds + x with the right-hand side
has the solution Bs+ !(x|co). Hence, by Theorem 2 and by (20),
(s+l)(xcD + y)sdx = B's+1(y\cD)
for any two p-adic numbers co ^= 0 and y. A second solution of Ds + x is
Fs+1(x|co); therefore, by Theorem 1, £s+ ^xlco) — Fs + ^xlco) is
independent of x so that
B'a+i(x\a>) = F's+l(x\co) = (s+ l)Ba(x\(o). (25)
It follows that
(xcd + yfdx = Bs(y\co) = cosBs
y_
CO
and
x"dx = B.
(s = 0, 1, 2,.. .). (26)
The second formula allows us to determine the integral of any
polynomial in x..
It is clear that the integral depends linearly on the integrand so that
(G(x) + G*(x))dx =
cG(x)dx = c
G(x)dx +
G(x)dx
G* (x) dx and
for any two functions G and G* and for any p-adic constant c.
Therefore, if
P(x) = P0 + PlX + ... + Prxr
is an arbitrary polynomial with coefficients in Qp, then by the second
formula (26)
P(x)dx = B0P0 + B1P1 + ... + BrPr.

288 Definite integrals and difference equations
Now, by the binomial theorem,
(xa> + yy= t [k )vkxkys-k
and therefore, by (26),
5
S
f*
s
S
Bs(y\co) = £ L ) ™kys"k xk*x = I (J BkVkys~k
h
(5 = 0,1,2,...) (27)
whence, for cd = 1,
*.(*)= El)5**""* (5 = 0,1,2,...). (28)
This formula gives an explicit expression for the Bernoulli polynomial
in terms of the Bernoulli numbers.
9 The denominator of B2n
We require an upper estimate for the p-adic value of Bs and for this
purpose shall prove a famous property of the Bernoulli numbers.
However, we first note that for s > 2, by the difference equation Ds
and by the second identity (24),
Bs = Bs(0) = Bs(l) = (- 1)-5,(0) = (- im
and therefore
Bc = 0 if s > 3 is odd.
Hence it suffices to study Bs for even s>2.
For seJ and for any positive integer N put
k = 0
so that
N- 1
S0(N) = N and SS(N) = £ fcs for s>l.
fe= l
As r runs over J, by the definition of the integral and by the second
formula (26),
' Ss(f) ~
lim
r -*• oo P
xs dx = Bs (s = 0, 1, 2,.. .)• (29)
'p

The denominator of B2n
289
We next add some inequalities. From the definition of SS(N),
pr + 1 _ l pr- l p- I
ss(f+1)= E ** = Z Z a+JfY
k=0 i=0 j=0
s pr- 1 p- 1 / \
= z z z (?W-*/
and therefore
fc = o i = o j = a \^
5^^) = ^^)^^-/.(^)5^).
Here, for every positive integer n,
S0(P) = P and ( ^)^1(^) = ^(^- 1)P
are both divisible by p. Therefore
S2n(pr +1) = pS2n(pr) + terms divisible by pr + l,
whence
S2n(Pr + l) S2n(p')
s + 1
<1.
p p
Let now w be any positive integer. Apply this inequality for r = 1,
2,..., u — 1 and add the results. It follows that
S2n(Pu) S2n(p)
P P
and so, by (29), as u tends to infinity
S2„(P)
<1.
B2n~
<1.
(30)
If, firstly, p = 2, then
S2„(2)=l=-1 (mod 2),
so that for all n > 1,
|B2„+il2<i- (31)
Secondly, let the prime p be at least 3. The following two theorems
are proved in elementary number theory.
(a) If the polynomial P(x\ of degree r, has rational integral coefficients
not all divisible by p, then the congruence
P(t) = 0 (mod p)
has at most r incongruent roots (mod p).

290 Definite integrals and difference equations
(b) Each of the two congruences
p^t) = (t- 1)(/: - 2).. . (t - p + 1) = 0 (mod p) and
p2(t)=tp~1 -1 = 0 (modp)
has the same p—\ incongruent roots /:=1, 2,...,/?—1 (mod p\
Thus the difference polynomial
p{t)=p1(t)-p2(t)
is at most of degree p — 2, but the congruence
P(t) = 0 (mod p)
still has the p — 1 incongruent roots t = 1,2,..., p — 1 (mod /?). Hence
all the coefficients of P(t) and therefore also all the coefficients of its
derivative P'(t) are divisible by p.
Now the logarithmic derivative of Pl (t) can be written as
Fl(t) = V 1 ='y(t(i ^-1
and so can be expanded into the formal power series in powers oft'1.
pr /f\ p — 1 oo oo
-¾ = x z fc»t-o- + i> = (p_1)t-i+ £ sjrtr*'"-1)
fll'J fc=lm=0 m=l
Similarly,
Pi(t) (p-l)f-2 p-l
( = 0
— = (p-i) y r1_,(p_1)
p2(t) t""1-! ((l-r^-1*) ^ ' L
Further
P[(t) P'2(t) = P'2(t) + P'(t) P'2(t) _ P'(t)P2(t) - P(t)P'2(t)
Pi(t) P2(t) Pi(t) + P(t) P2(t) Pi(t)P2(t)
where all coefficients of the polynomial in the numerator are divisible
by /?, while the denominator has the highest coefficient 1 and has
rational integral coefficients. Therefore the coefficients of all the
negative powers of t in
00 \ oo
m= 1 / / = 0
are likewise divisible by p, and so it follows that in particular,
s ( ^ = 1-1 (mod^ if P-1^
2nW~1 0 (mod p) if p-\\2n.

The case of an analytic function 291
By (30), this means that
1
*U + P
<1 if p-l\2n, \B2\<1 if p-1/211,
a result which by (31) remains valid also for p = 2 except that then
p—l = l always is a divisor of 2n. We have thus proved a famous
theorem by Clausen (1840) and von Staudt (1840).
Theorem 5 Let 2n be a positive even integer, and ktpl9p2,... ,pv
be all the distinct primes for which p — 1 divides 2n. Then there is a
rational integer A2n such that the Bernoulli number
/11 1
B2h = A2h-[ — + — + ...+-
\Pi Pi Vv,
By way of example,
B24=-86579-(i + i + i + 7 + T3)-
I am indebted for this proof to J. W. S. Cassels (private
communication).
Theorem 5 has the important consequence that for all primes p and
all suffixes s,
\BS\P<P (32)
because this inequality trivially holds when s = 0, s = 1, or s > 3 is
odd. Hence, by (27),
\Bs(x\(d)\p <p if both x and co lie in Ip. (33)
10 The case of an analytic function (7(x)
Assume that
00
G(x)= £ Cmxm, where lim Cm = 0,
m = 0 m-*co
is an analytic function on Ip, and that both y and co lie in Ip. We found
that
/•
(cox + y)sdx = Bs(y\co) (s = 0, 1, 2,...).

292
Definite integrals and difference equations
Hence it follows from the power series for G(x) and from (33) that
00
G(cox + y)dx = £ CmBm(y\co\
m = 0
(34)
and in particular,
00
G(x)dx±= £ CmK
m = 0
It is obvious that the infinite series on the right-hand sides of these
integral formulae are convergent.
It is also possible to determine an explicit function F(x) which
satisfies the difference equation
F(x + co)-F(x)
D : = G(x),
(JO
provided that the hypothesis Cm->0 is replaced by the stronger
assumption that
lim m\CJp = 0.
(35)
m-+ oo
We found that the special difference equation
F(x + cd)-F(x)
= xr
CD
(5 = 0,1,2,...)
has the solution
It follows therefore that the difference equation D is satisfied by the
infinite series
00
F°(x\co) = ^ Cm(m+l)-1Bm+1(x\co),
(36)
m = 0
which by (35) is convergent under our restrictions on x and co. In this
series, all the terms are normed by the identities (21). It follows
therefore that also F° is normed, at least if AT is again restricted by the
condition that pfN. Thus
N- 1
IH* +
kw
k = 0
N
co\=NF0{x
CO
jj) if PiN9
and F°(x -co\-co) =F°{x\co). (37)
We may in (36) replace the Bernoulli polynomials by their explicit

A second sequence of difference equations 293
expressions as polynomials in x and obtain then a power series for F°
which is easily seen to be convergent on I Hence F° is itself analytic
on I If the series for F° is differentiated with respect to x and
afterwards x is replaced by y9 Theorem 2 leads to another proof of the
integral formula (34), but under the more restrictive condition (35) for
the coefficients Cm.
11 A second sequence of difference equations
Our first sequence of difference equations Ds corresponded to
functions Gs(x) that were polynomials in x. The second sequence will
correspond to functions two of which involve the p-adic logarithm L,
while the others are rational in x.
The variable x is restricted to Ip, while X and co denote two
parameters such that
\X\p<l/p and |co|<l.
The logarithmic function L(a) was for \a — 1| < 1/p defined by
oo / I \m — 1
L(a) = I y—^—(° ~ !)m
m= 1 m
In terms of this function put
G_ y(x\X) = (1 + Xx)L(\ + Xx) - /be, G0(x\X) = L(l + /be),
Gn(x\X) = (1 + Xx)~n (n = 1, 2, 3,.. .)•
All these functions are analytic on lp, with the convergent power
series
oo ( \\m
m
v-, (- 1)
g_1(x\x)= yy—tt-X"*
1V ' } ^2(m-l)m
m
oo / 1 \m — 1
v, ("I)
G0(x|/l)= y Xmx
00 ' -n.
,m
m = 0\ m J
(n = 1, 2, 3,...). (38)
On differentiating these functions, we find that
G'_ ^1/1) = XG0(x\X\ G'0(x\X) = XG^xlX),
G'n(x\X)=—nXGn+l(x\X) for n>\.

294
Definite integrals and difference equations
It is obvious that the power series (38) have coefficients which
satisfy the condition (35). This allows us to evaluate the integrals of
the functions; by (34), for yelp,
00
G_l(a>x + y\X)dx= £ / ;
mf 2 (m - l)m
^mBm{y\co),
00
J/,
(- If1"1
G0(cDx + y\t)dx= X ^ '- ^Bm{y\(D\
m
m = 1
oo
Gn(cox + j|A)dx = Yj
m= 0
— ft
m
^^m(y|co)
(ft = 1, 2, 3,...). (39)
Again all the series on the right-hand side are convergent.
Next, associate with each function Gn(x\X) the difference equation
F(x + co) - F(x)
A- : - - — = GH(x\X) (ft = - 1, 0, 1,...).
CO
By means of (36) we obtain then the following sequence of normed
solutions F°(x|co,/l) of An,
oo / I \m 2 m
F°_ ^XICO, X) = ^ —. 1^./-.. , ^Bm+l(X\C0)>
m
= 2(m— l)ra(ra-f 1)
oo / 1 yn — 1 t m
F%(x\(o9X)= Y, „./„. , 1. Bm+i(*|a>)>
m
oo
= i fti(m + 1)
,- l
— ft
AmBm+1(x|co)
F°(x|cM) = ^ (w+1)' ,
m = 0 \ m
(ft = 1, 2, 3,...). (40)
All these series are convergent and can easily be reordered into
power series which converge on I \ hence they represent functions
analytic on I
Since the polynomials Bm + i(x\co) are normed, the same is true for
the functions F°(x\co, X). Thus for all suffixes n = — 1, 0, 1,...,
N~'J /ceo
I F°n(x +
k = 0 \
N
<D,n=NF°[x
CO
N
a) for pXN,
F„(x — co| — co, /I) = F°(x|co, X).
By (27), the Bernoulli polynomials have the homogeneity property
Bs(tx\t(jo) = tsB(x\(D).

The special basic function F0(x\X) 295
It follows therefore from (40) that
F°n(tx\tco,X) = tF°n(x\co,tX) (n=-l,0,l,...), (41)
whence, for t = co~ 1, and with X replaced by coX,
x
CO
and so the identities for F° may be written as
k
F°n(x\co,X) = coF°n
1, coX \ (n = — 1, 0, 1,...),
1,A) for pIN,
N
19X) = NF°[Nx
k=0
F°n(l -x\l,X) = ~F°n(x\l -4 (42)
It would thus suffice to consider the special case when co = 1.
To these formulae we may add the following ones for the
derivatives with respect to x; they are a consequence of (39) and can
also be obtained from the series for the functions F° by
differentiating the Bernoulli functions,
F2\(x\co, X) = XF°0(x\co, X),
F°'(x\cq,X)^XF°1(x\cq,X),
F°n\x\co, X)=- nXF°n+ x(x|co, X) (n = 1, 2, 3,...). (43)
The most interesting one of these functions is Fq(x\co, X) which we
shall now study in a little more in detail.
12 The special basic function F0 (x\ A)
By the homogeneity relation (41) there is no loss of generality in
assuming that the parameter co has the value co = 1. Instead of the
normed function Fq(x|1, X) we shall deal with the function
F°o(x\l,X)-F°o(0\l,X) = Fo(x\X\ say,
which vanishes for x = 0, hence in the earlier notation is basic. It may
by (40) be defined by the convergent series
oo / i\m— lorn
^=^-^^^^-^11 (44)
and it satisfies the difference equation
F0(x + 1\X) - F0(x\X) = L(l + Xx).

296 Definite integrals and difference equations
Hence, if n is any positive integer,
V(F0(/c+l|A)-F0(/c|A)) = F0(n|A) = Lrn\l + /cA)\ (45)
fc= 0 \fc =0 /
Apart from the hypothesis that \X\P <p~\ there is no further
restriction on X. We may therefore choose
k = Vlh
where j is any one of the integers 1, 2, ..., p — 1.
On substituting these values for X in (45) and adding over them, it
follows that
p-i
I F0[n
J/ \ j = 1 fc = 0 /
Here, as is proved in elementary number theory,
(P -1)1=-1 (mod p),
hence
|(p-l)!2-l|p<l/p.
Hence the logarithm L((p — 1) !2) is defined, and we may write the last
identity in the form
2l(PU "rf (/ + M) =2¾1 ^of^ t) + nL((p - l)!2). (46)
\ j = 1 * = o / j = 1 \ i /
We introduce now the p-adic gamma function as defined by Yasuo
Morita (1975). If u is any positive integer, put
J»= n t;
1 < t < u - 1
(p, t) = 1
if u = 1, the product is empty and is to mean 1. In this notation,
n "n (/+m=rpim\
j = 1 fc = 0
so that (46) is equivalent to
L(/>p)) = Y F0 ( n
j) + Imp - 1) !2). (47)
If in this equation n is replaced on the right-hand side by a variable
x in / it becomes an analytic function on Ip. Hence L(rp(px)) is by
this formula defined as an analytic function on Ip. Here \px\p < 1/p.

Properties of the Bernoulli function 297
For further properties of this analytic function refer to Morita's (1975)
paper and to recent work by Gross and Koblitz (unpublished).
13 The Bernoulli function
To conclude this chapter, let us derive from tfie Bernoulli polynomial
Bs(x\cd) an analytic function of s. For this purpose it is necessary to
impose on x and co the restrictions
\x-l\p<l/p, hence [x|p = l, and \co\p<l/p. (48)
By this restriction on x, the function
' S
F(x;s)= X (x-l)w
n= 0 xH
of Chapter 14 § 4, is defined for all selp and is continuous on Ip. Since
this function has the value xs when x e J, we shall from now on write xs
for F(x; s) even when s does not lie in J, but belongs to Ip.
As long as s is a non-negative rational integer, Bs(x\co) can be
written as the seemingly infinite series
'S
Bs(x\co)=xsYJ Bk{cD/xfl\ (49)
which in fact breaks off since its terms with k> s are equal to 0. In this
representation, by the hypothesis (48),
\Bk (co/xf\p<p1~k^0 as fc->oo.
Hence, if x and a> satisfy (48), the series (49) converges and represents a
continuous function of 5 for all s in Ip. For p > 3 the function of s
defined by (49) is in fact analytic if s elp, and this remains also true for
p = 2 if the right-hand side upper bound 1/p is replaced by \.
We call Bs(x\cd) the Bernoulli function for general selp. The
restriction (48) on x excludes the value x = 0; it is thus not possible by
this method to extend the Bernoulli numbers Bs to general suffixes s in
14 Properties of the Bernoulli function
For seJ the homogeneous polynomial
Bs(x\o))= f (°)Bko>kx°-k

298 Definite integrals and difference equations
was found to have the following properties.
Bs(x + co\co)-Bs(x\co) _^s_1^
CD
(B) B's(x\co) = sBs_1(x\co),
where as usual the dash denotes differentiation with respect to x; and
N~l ( km
, = o v N
co = NB„ x
CO
N
and Bs(x — co\ — co) = Bs(x\co).
Here the integer N may be divisible by p.
We show now that for general s in Ip the Bernoulli function Bs(x\co)
retains these properties, except that now p must not be a factor of N.
The proof will be the same for all these formulae. It suffices to show
that all the functions that occur in (A), (B), and (C), are continuous
functions of selp. For these relations hold for sg J, and J is dense in
In the equation (A), by (48), also
\(x + co)-l\p<l/p.
Therefore not only Bs(x\co) and sxs_1, but also Bs(x -\-co\co), are
continuous functions of se/ .
In the equation (B), the development (49) implies that, at least for
sg J,
00 / c
B's(x\co) = xs-1YjBk(s-k)(co/x)
k = o \k
since the terms with k> s are again equal to 0. If, however, s is in I
then
lim Bk (s - k)(co/x)k = 0,
k-+ oo
and so B's(x\co) can be continued to a continuous function of selp. The
same is true for the other function sBs_ !(x\cq) in (B).
Finally, in (C), assume that p\N. Then, by the hypothesis (48),
|(x + (fco)/iV))-l|p<l/p for fc = 0,1,..., N-1;
\co/N\p<l/p; and |(x -co) - l|p< 1/p.
This implies that all the functions occurring in (C) depend
continuously on selp. This concludes the proof.

Problems
299
Without any restrictions on oo and y we had proved that for se J,
(cox + y)s dx = Bs(y\co). (50)
If it is now assumed that
\y-l\p<l/p and \co\p<l/p,
then
\(cox + y)-l\p<l/p for xe/p.
Hence the integrand in (50) is now defined for all s in Ip9 and by
continuity this formula remains valid also for sel . However, we are
not allowed to put oo = 1 and y = 0, and the former equation
xs dx = Bs
cannot be extended to the case when 5 lies in I and not in J.
15 Problems
1 Use the identity
) = ( ) + ( ) to prove for ne J that
72+1/ W+l/ W
n n+1
and determine the more general integral
x+y
dx.
n
2 Determine
axdx if \a — 1L < -.
3 Fl9 F2, Gl9 G2 are four continuous functions tp-^Qp such that
Fj(x +1)- F/x) = G/x) (j = 1 or 2),
and that F1(x)F2(x) is differentiable at x = 0. Show that
J/,
Fx(x+l)G2(x)dx+ F2(x)G1(x)dx = (F1(x)F2(x))'x = 0.
Ji.

300 Definite integrals and difference equations
4 Determine the integral
((x + l)n-xn)£s(x + l)dx
where n and s are positive integers.
5 The following proof evidently is false; decide where the error is. 'By
the formulae (4) and (5), if G(x) = (%^) and F(x\co) = ( X^ \
then
F(x + oo\oo) — F(x\co)
= G(x).
00
Therefore, by (10), there exists a p-adic quantity B(oo) such that
F(x — co\ — go) = F(x\oo) + B(oo\
identically in x, that is
n+1 J \n+\)
Assuming that the integer n is at least 2 and putting x = 0, it follows
that B(oo) = 0. But it is clearly not true that
1 — (x/co)\ ( x/oo
n+1 ) [n+1
identically in x/oo'
6 Let G:In-^Qn be continuous on /„, and 0<|coL<l, and let
p y^p p-> i \p — ?
further
00 fx/oo\
G(x)= X b„A for xeooIp.
n=0 \ H J
Determine the most general continuous function H : ooIp -> Qp
satisfying
H(x + co) + H(x) = 2G(x).
7 X(x) is the characteristic function of a ball |x — a\p<p~s in I
Determine
X(x)dx.

18
Functions on the quadratic extension
fields of Qp
Up to now, we have studied functions that were defined on I or on
some subset of Ip, in particular J. A simple change of variable allows
us to extend the results that were obtained to functions on more
general subsets of Qp.
It is of great interest to generalise the theory to functions defined on
subsets of algebraic extension fields of Qp. For analytic functions
defined by power series there is a vast literature dealing with many
interesting functions. Rather less is known for continuous functions.
Of particular interest are again the functions that can be defined by
convergent interpolation series. In the present chapter we shall
investigate the functions which can be defined by an interpolation
series which converges on the set of all integers of a quadratic extension
field of Qp. For the analogous problems for arbitrary finite algebraic
extension fields of Qp see my paper (Mahler 1975).
1 The quadratic extensions of Q
In Chapter 6 we obtained all the distinct quadratic extension fields
Kp = Qp(<sfd) of the p-adic field Qp. The result was that there were for
p = 2 the seven fields
and for p > 3 the three fields
QPUNp\ Qp(Vp)> Qp(-JWp);
here Np is the smallest quadratic non-residue (mod p). We denoted by
K% the field Q2{-J-^) and put
fi(_l+>/T3) if Kp = K*2,
J \/d if KP + K*.

302 Functions on the quadratic extension fields
Every element z of Kp had a unique representation z = x + y^fd
where x and y lie in Qp. With z was associated the real non-negative
number \z\ defined by
\z\p=+(\x2-dy2\p)1'2,
which coincides with the p-adic value |x| when z = x + 0^/d lies in
Qp. It was shown that \z\p is a valuation also on Kp and that K is
complete with respect to this valuation.
The elements z of Kp such that
were called the Kp-integers\ they formed a subring /^ of Kp, and
zeKj, was such a Xp-integer if and only if it could be written in the
form ,-1. r
z = x+ jy, where x, y elp.
The 7 + 3 possible quadratic extensions Kp of Qp could be
subdivided into the 8 ramified fields
(R) QiiV^l 62(n/2), 62(73¾ Q2(n/% 62(/^6),
22(x/6X 6,(7¾ QP(\/p*0
and the 2 unramified fields
(u) 62(/1¾ e„(7v
In each of the ramified fields Kp the prime /7 split into a product
P = EP\
where E and P were j^-integers with the valuations
\E\p=l and \P\p = p~1/\
This had the consequence that every Kp-integer z had the canonic
expansion
Z = Zq + ZyP + Z2P2 + . . .
in powers of P rather than of p\ here z0, zl9 z2,. . . were digits 0,
1,..., p — 1. lfzeKp was not a j^-integer, then finitely many terms in
negative powers of P had to be added.
If, however, Kp was one of the unramified fields (U), then p
remained a prime in Kp, and there was no element P of Kp such that
\P\p = p~1/2. In this case, every j^-integer z allowed a canonic
expansion
Z = Zq + ZyP + Z2P2 +...,

Analytic and regular functions on IK 303
where Z0 = x0 + jy0, Z1=x1+ jyl9 Z2=x2+ jy2,. . . are new digits
of the form
r + js, where r, s = 0, 1,.. ., p — 1.
If zeKp was not a j^-integer, then again finitely many terms in
negative powers of p had to be added.
2 Analytic and regular functions on IK
Since each quadratic extension Kp is a field with the valuation | z | , it is
also a metric space with the distance d(z,w) = \z— w\p. Hence all the
definitions and results of Chapter 7 on such spaces can be applied. In
particular, IK can be shown to be a compact subset of Kp.
We shall be concerned only with functions F : IK -> Kp. In the same
way as z = x + jyeIK, where x and y lie in I can be split into a 'real'
part x and an 'imaginary' part y, so the function value F(z)eKp allows
a decomposition
F(z) = u(x, y) + >(x, y\ where w(x, y), t;(x, y)eQp,
into a 'real' part u(x, y) and an 'imaginary' part v(x, y). A general
function F on /^ is thus essentially a pair (w(x, y\ v(x, y)) of two
functions
u:IpxIp-^Qp and v:IpxIp-^Qp
of two variables, each on /
We are not interested in such general functions, or even in functions
that are continuous and therefore uniformly continuous on IK, but
only in the two special classes of functions defined as follows.
Definition 1 F :IK-^Kp is said to be analytic on IK if it can
be written as a power series
00
F(z) = ^ a„z"
n = 0
which converges for all z on IK.
Definition 2 F :IK-^Kp is called regular onIK if it can be written

304 Functions on the quadratic extension fields
as an interpolation series
which converges for all z on IK,
Here in both representations the coefficients an and An may be any
elements of Kp.
In the case of an analytic function, the power series may be
differentiated term by term any number of times, and the coefficients
an can be expressed in terms of the successive derivatives of F(z) at
z = 0by
an = — (72 = 0,1,2,...). (1)
n\
The function F is thus completely determined by its values in an
arbitrarily small neighbourhood of z = 0. All this is well known and
can be found in any one of the books dealing with p-adic analytic
functions, e.g. in Hensel (1913).
For regular functions on IK only a possibly weaker statement can
be made. The set J = {0,1,2,...} is a subset of I and hence also of IK.
If now F is regular on IK, its interpolation series holds in particular at
all the points of J, and therefore its coefficients An are defined by the
usual law
A»= i0{~lT~k(j)m (n=0,l2'--x
where, however, now the function values F(k) and hence also the
coefficients An lie in Kp, but not in general in Qp.
Since the interpolation series converges on IK and hence also on the
subset I F is continuous on I and so has the property (N),
lim |y4n|p = 0.
n -*• oo
This limit formula will soon be replaced by a stronger one. We shall
find that also a regular function has everywhere on IK derivatives of
all orders and so is continuous on IK. In the proof, we shall have to
distinguish whether the extension field Kp is ramified over Qp or not,
and the result will be particularly simple for the two unramified fields
Q2(V^3) and Qp(Jn~p).

Unramified fields
305
3 Estimates on the unramified fields
In the two unramified fields Q2w~—3) and Qp{y/Np) the number p
remains a prime, but instead of the old digits 0,1,. .., p — 1 there are
now the p2 new digits
r + sj, where r, s = 0, 1,..., p — 1.
In particular; the new digit j = 0 + lj lies in Xp and /^, but not in Q
Ip, or J, and it has the obvious property that
\j — wlp = 1 f°r we«J- (2)
It is convenient to put
(z, 0) = 1 and (z, n) = n\ ( J = z(z — 1).. .(z — n + 1)
for n = 1, 2, 3,.. .
Then, from (2),
I0»lp=1 for ^eJ. (3)
On the other hand, if z is a i^-integer, the same is true for z — n if ne J,
and it follows that also (z, rc) is a j^-integer, i.e.
| (z, n) \p < 1 for rc e J. (4)
For the binomial coefficients I ) and I ) the formulae (3) and (4)
w w
imply that
n
1
n\
for neJ,
(5)
and
72
<
1
n
for zgL and neJ.
K
(6)
Assume now that F is regular on IK. Then its interpolation series
converges in particular for z = j, and hence its general term An\
tends to 0 p-adically. This means by (5) that
A.
n
lim
n -*• oo
n!
= 0.
(7)
Conversely, if this relation (7) is satisfied, then it follows from (6) that

306 Functions on the quadratic extension fields
the interpolation series for F converges also at the arbitrary point
zeIK. Hence the limit relation (7) is for unramified fields Kp the
necessary and sufficient condition for F to be regular on IK.
Now, in Theorem 4 of Chapter 14 it was proved that (7) is also the
necessary and sufficient condition for F to be analytic. The proof was
then for functions of I but it is clear that it carries over to the present
case without change.
Hence for the unramified fields there is no distinction between regular
and analytic functions. In particular, a regular function has derivatives
of all orders everywhere on IK.
4 Estimates for ramified fields
A very different result will be found for the 8 ramified fields (R). In
these fields p is no longer a prime, but has a factorisation
p = EP2, where \E\p=l and |P|p = p"1/2.
The K -integers z are still of the form
z = x+jy, where x,yelp.
But now j = P except for the two fields Q2(\/ — 1) and 62(%/3) f°r
which j = P — 1. The canonic expansion of every j^-integer z has the
form
z = z0 + z1P + z2P2 + ...,
where the coefficients z0, zl9 z2,.. . are old digits 0, 1,..., p — 1.
From this canonic series it follows at once that for every neJ,
\z — n\p< \P\p < 1 if n = z0 (modp\
but that
I z — n\p = 1 otherwise.
The integers ne J satisfying n = z0 (mod p) form again an arithmetic
progression of difference p. It follows therefore that for all zeIk,
1(2,11)1^1^1^-1 (11 = 0,1,2,...). (8)
In the special case when z = P this estimate can be replaced by the
exact formula
\(P,n)\p = \P\W* (n = 0,1,2,...). (9)
For now | P — n\ is equal to 1 if p does not divide n, and is equal to | P\p
if p is a divisor of n because then \n\p < \p\p < \P\p.

The addition formula
307
Let now F(z) be a regular function on IK; its interpolation series
converges then at all points of IK. In particular, it converges at z = P,
and here its terms tend to 0. Now, by (9),
n
= \n\\-'\(P,n)\=\n\\-'\P\
[n/p]
\p IV- 5 -vip \--\p I- 1/7
Hence
lim
n -»■ oo
,4
n!
!Lp[«AP]
= 0.
(10)
By (8), this limit relation implies also that F(z) converges at arbitrary
points z of IK. Hence the following result has been proved.
Theorem 1 The function F :IK-^Kpis regular on IK if and only
if its interpolation coefficients An satisfy the equation (10).
By |P|p < 1 this equation (10) is weaker than the condition (7) for F to
be analytic. Hence functions on ramified fields may be regular without
being analytic. A simple example is given by the function
00
F(z)= £ n?p-[»/p] +nog"]
n = 0 \n
which satisfies (10), but not (7).
5 The addition formula for regular functions on ramified
fields
If the function F : IK -> Kp is regular on IK9 then its coefficients can by
(10) be written in the form
AH = nlP-M*eH (.2 = 0,1,2,...), (11)
where {en} is a null sequence in Kp.
Denote now by z and w two variables in IK and put
z \ l wv
Bmn — ^
m + n
m \n
(m, n = 0, 1, 2,...).
By (11),
Bmn = m!P- W"| M x nlP-^r ) x £m„
(m, n = 0, 1, 2,...),

308
Functions on the quadratic extension fields
where Emn denotes the number
£ _ I m + n \ plm/p] + [n/p] - [(m + n)/p]e
m
'mn
m + n
(m, n = 0, 1, 2,. . .).
Here, by (8),
m
\p~ [m/p]
m
< \P\~ 1 and
n\p-Wp]
w
n
<|P|_1
and it is further obvious that
m + n
m
< 1 and [m/p] + [n/p] — [(m + n)/p] > — 1.
Hence the first two factors of Bmn are bounded for all m and n, and the
last factor Emn tends to 0 as m + n tends to infinity. Hence
lim Bmn = 0. (12)
m + n -* oo
This equation proves that the double series
00 00
A= Z Z Bm„
m = 0 n = 0
converges uniformly for (z, w)elp x /r Since the field Xp is non-
archimedean, the terms of this double series may be rearranged in
any order.
First add together all terms of A for which m + n is equal to a given
integer r > 0. Since
z (*)("
-n = rvnj\n/
z + w
m +
m > 0
n> 0
it follows on summing over r that
z + w
00
A= Z 4
= F(z + w).
r= 0
Secondly, the partial sums
00
00
2j ^rnn ~ 2j w + n'
m = 0
m = 0
W
m \ n
are all convergent, and they tend by (12) to 0 as n tends to infinity.

Hence, on putting
> z
An(z)= X Am + n[ ) (,2 = 0,1,2,...), (13)
m = 0 V
it follows that A has also the value
00
A= E 4.00
n = 0
We obtain therefore the addition formula
00
w
F(z + w)= £ An(z)[ . (14)
n=0 W
When z and w are restricted to I we come back to a formula proved
for all continuous functions on I
6 The series A'n
In much the same way as in Chapter 13 § 6, put
oo / i \m — 1
A'H= £ V '- Am + „ (/1 = 0,1,2,...). (15)
m=l W
By (11),
-^m + B = n!F-^em + Bx(m + n)(m-l)!P-^
m \ n J
x p[m/p] + [n/p] - [(m + n)/p].
It was already remarked that the p-adic value of is at most 1
\ m J
and that [m/p] + [n/p] — [(m + n)/p'] is never less than — 1. Further,
as was used before,
I r M = n " ^ " ^2] ~ fr/P3] ~ • • •
r • l/> F
Therefore the two sequences
{(m-l)!P"[m/;,]} and {rc!P-[n/"]},
in which m runs from 1 to infinity and n from 0 to infinity, are both
bounded. Therefore a positive constant c independent of m and n
exists such that
1
-A.
™ m +n
m
for all suffixes m > 1 and n > 0. (16)

310
Functions on the quadratic extension fields
This property shows already that all the series (15) are convergent.
It further follows from (15) and (16) that
\A'\ < r\niP~Wp]p I
hence that
lim
n -*■ oo
AL
n\
lp[n/p]
= 0.
This means, by Theorem 1, that the new function F': IK
by
Kp defined
00
F'(z) = I K
n = 0
n
is again regular on I
K'
7 The relation between F(z) and F(z)
We next study the relation of F'(z) to the original function F(z). By (8)
and (16),
1
-A
m \n
<c
n\p- \.nIP\p v pln/P] ~ 1
n .r cm + n a r
n\
— C\P em + n\v'
Here em + n tends to 0 as m + n tends to infinity. Hence the new double
series
\m — 1
oo oo / 1Y- - I ~
v= E I ^-^-^
m = 1 n = 0
m
Lm + n
ft
converges uniformly for all z on /^. On summing first over m and then
over ft, we find that
V = F(z).
If, on the other hand, we sum first over n and then over m, the result is
that
00
m= 1 m
Therefore
00
(- If"1
F'(z) = E 4..(4
(17)
Theorem 1 of Chapter 13 suggests at this point that F'(z) is in fact

The relation between F(z) and F'(z)
311
the derivative of F(z) at the point z. Here, for functions on IK, the
derivative F(1)(z) of F(z) at z is again defined as the p-adic limit
0<|wL-0
This limit can be evaluated as follows. By the definition (13) of An(z)
and the addition formula (14),
F(z + w) - F(z) 1 " , VhA » An(z) w-i
n= 1
00 00 A
= Z I
m = 0 n = 1
n= 1
m + Mi z\/w-l
n \m/\ ?i — 1
(18)
Here, by estimating in a similar way to before,
^m + n ' Z
n
w — 1
m/ Vn— 1
<c
(m + n)
n
p- [(m + n)/p]e
m + n
1 1
X __p[»»/p] " 1 p[(w " D/p] " 1
m! (fi — 1)!
= c
m + n
p[m/p] + [(n - l)/p] - [(m + n)/p] - 2g
because
<c\P
m
4„
m + n
^m + nip'
[(« - l)/p] > [»/p] -1
and therefore
[m/p] + [(n - l)/p] - [(m + n)/p] > - 2.
This shows that the terms of
fw-i
m)\ n—\
00 00
m=0n=l
Am + n ( Z
n
tend to 0 uniformly in z and w as m + n tends to infinity.
In order to determine F(1) (z) we may then allow w to tend to zero in
each term of this double sum. It follows then from (17) and (18) that
F^\z) = F'{z\ (19)
leading to the following result.
Theorem 2 If Kp is ramified, then every regular function on IK
has a first derivative which is again a regular function. It has
therefore regular derivatives of all orders.

312 Functions on the quadratic extension fields
However, since a regular function on IK is not in general analytic, the
formal Taylor series
n = 0 f * •
need not converge for all z on IK. It does not seem to be known
whether this power series converges on some subset \z\p < p~s of IK,
nor whether F(z) can vanish on a dense subset of IK without being
identically 0.
8 Partial differential equations
From what has been said in this chapter, both the analytic and the
regular functions on IK have some properties which are analogous to
those of analytic functions of a complex variable.
The analogy goes even further. As is well known, the real and the
imaginary parts of a complex analytic function satisfy the
Cauchy-Riemann and the potential differential equations. As we
show now, similar differential equations hold for the 'real' part u(x, y)
and the 'imaginary' part v(x, y) of an analytic or regular function
F(z) = u(x,y) + jv(x,y);
here
z = x + jy,
and j is in general equal to yfd except for the one field e2(v^3)
where; =^(- 1+^/-3).
Both in the analytic and the regular case F(z) has derivatives of all
orders. In forming these derivatives, we are allowed to change only
the 'real' part x or only the 'imaginary' part y of z; the obtained
derivative will be the same in either case. This implies immediately
that u(x, y) and v(x, y) have partial derivatives relative to x and y of all
orders.
Consider, say, the first derivative F'(z) of F at the point z.
Depending on whether jcorj; are changed, it follows that
dF(z) du(x, y) dv(x, y)
-jz-=F{z)=^x-+J~^r

Partial differential equations 313
and
= F (Z)J = —^ +J-
Hence
dy dy dy
du(x,y) .dv(x,y) du(x, y) . dv(x,y)
+ J—^ = —z J + —; J -
dy dy dx dx
In this identity both the 'real' parts and the 'imaginary' parts are the
same on both sides, and if K = Qp(^/d) is distinct from the special
field K* = Q2(\/— 3), then j =y/d, while in the excluded case
7=^(-1+^-3) and therefore j2 =^(- 1 +V~ 3)- Hence for
Kp ^= ICf the 'real' and 'imaginary' parts of F(z) satisfy the partial
differential equations
du(x, y) = dv(x, y) ^ du(x, y) = d dv(x, y)
dx dy dy dx
except in the case Kp = K% when the differential equations take the
form
du(x, y) = dv(x, y) [ dv(x, y) ^ du(x,y) [ dv(x,y) = Q
dx dx dy dy dx
The similarity of these equations to the Cauchy-Riemann differential
equations for complex analytic functions is striking.
From (20) and (21) one of the two functions u and v can be
eliminated. Then, for Kp ^= K%, we obtain the formulae
dMy)J2u(y) and dmdMy)J2v(x9y)m (22)
dx2 dy2 dx2 dy2 y }
If, however, Kp = K%, the result becomes
d2u(x,y) d2u(x,y) d2u(x,y) =
dx2 dxdy dy2
and
d2v(x, y) d2v(x,y) d2v(x,y) =
dx2 dxdy dy2
These differential equations are analogous to the equation of the
logarithmic potential.
All these equations have been found to be necessary for F to be

314 Functions on the quadratic extension fields
analytic or regular. It remains an open question to what degree they
are also sufficient.
9 An example
Let Kp be the special field Q2{\/2) so that Kp is ramified, p = 2 = P2,
d = 2. We choose for F(z) the function
00 /z
F(z)= £ (-2)"
n = 0 \"
which for ze J is equal to (1 — 2)z = (— l)z, and we assert that for zeIK
this function is regular, but not analytic.
For neJ put
e{n) = ln/2] + [n/4] + [n/8] + . ..
so that
\n\\2 = 2-e(n\
If n = 2m is an integral power of 2, then
e(n) = 2m ~ 1 + 2m ~ 2 + .. . + 2 + 1 = 2m - 1,
hence
l(-2)7«!|2=i,
showing that {(— 2)n/n!} is not a 2-adic null sequence, and hence that
F is not analytic.
For general neJ and for p = 2, by Chapter 14 § 4,
in! 12 = 2"(n-{n}).
I I +4
Here
{rc} = 7i0 + nl + n2 + ... + nr if n = n0 + 2^ + 4w2 + ... + 2rnr5
the coefficients n0, nu n2, • •., nr being digits 0 or 1. It is clear that
{n} > 1 if ft > 1,
hence that for such n,
|rc!|2>2l~n-
It follows then that
(-2)"
n
(V2)[w/2]
<2x2-[n/2]<4x2"n/4
2

because
[n/2] >\n-2.
Hence, by Theorem 1, F(z) is regular on IK.
It is obvious that F(z) is a pseudo-constant if z is restricted to J or
to I2. I do not know whether it is also a pseudo-constant on IK.
10 Problems
1 The sequence {un} is defined by
u0 = 0, wx = 1, un + 2 = ?>un+l— un for ?2 > 0.
Decide whether {wn} can be continued to a continuous function on
the Xp-integers where Kp = Q5 (a/5), and if so, whether this function
is (i) analytic, or (ii) regular on IK, or (iii) neither.
2 Under which restrictions on a e IK is the function
n=0 W
(i) analytic, or (ii) regular on IK if Kp = Qp^Jd)^
3 Prove that for every positive integer n the nth derivative of a regular
function on IK is strictly differentiable on IK.
4 The function G:IK^Kp is regular on IK, and the function
F : IK -> Kp is continuous on IK, while both functions are connected
by the equation
F(z +1)- F(z) = G(z).
Prove that also F is regular on IK.
5 Is the function
F(z) = £ R! (Z)
n=0 \n/
(i) analytic, or (ii) regular on IK ?

References
Ahlswede, R. & Bojanic, R., 1975. Approximation of continuous functions in
p-adic analysis. J. Approximation Th. 15, 190-205.
Bachmann, G., 1964. Introduction to p-adic numbers and valuation theory
Academic Press, New York.
Bojanic, R., 1974. A simple proof of Mahler's theorem on the approximation of
functions of sl p-adic variable by polynomials, J. Number Th. 6, 412-15.
Clausen, Th., 1840. Theorem. Astronomische Nachrichten 17, 351.
Dieudonne, J., 1944. Sur les fonctions continues /?-adiques, Bull. Sci. Math. (2),
68, 79-95.
Hensel, K., 1913. Zahlentheorie. Goschen, Berlin and Leipzig.
Mahler, K., 1958. An interpolation series for continuous functions of a/?-adic
variable. J. reine angew. Math. 199, 23-34; 1961, Correction, 208, 70-2.
Mahler, K., 1961. Diophantine approximations. University of Notre Dame Press.
Mahler, K., 1975. On certain non-archimedean functions analogous to complex
analytic functions. Bull. Austr. Math. Soc. 14, 23-36.
Monna, A. F., 1970. Analyse non-archimedienne. Springer-Verlag, Berlin.
Morita, Y., 1975. Aj?-adic analogue of the iT-function, J, Fac. Sci, Univ. Tokyo,
Sec. IA, 255-66.
Miiller, H., 1975. Bemerkungen zur Approximation stetiger Fuhktionen in einer
p-adischen Variablen, Teil I. J. reine angew. Math. 276, 167-9.
Miiller, H., 1977. Bemerkungen iiber stetige Funktionen in einer p-adischen
Variablen. J. reine angew. Math. 290, 73-6.
Norlund, N.E., 1924. Vorlesungen uber Differenzenrechnung. Springer-Verlag,
Berlin.
O'Meara, O. T., 1963. Introduction to quadratic forms. Springer-Verlag, Berlin,
van Rooij, A. C. M. & Schikhof, W. H., 1971. Non-archimedean analysis. Nieuw
Archief voor Wiskunde (2), 19, 120-60.
von Staudt, K. G. Chr., 1840. Beweis eines Lehrsatzes die Bernoullischen Zahlen
betreffend. J. reine angew. Math. 21, 372-4.
Weisman, C. S., 1977. On p-adic differentiability. J. Number Th., 9, 79-86.
Further reading
Amice, Y., 1964. Interpolation /?-adique. Bull. Soc. Math. France 92, 117-80.
Iwasawa, K., 1972. Lectures on p-adic L-functions. Princeton University Press.
Koblitz, N., 1977. p-adic Numbers, p-adic Analysis, and Zeta-functions, Graduate
Texts in Mathematics. Springer-Verlag, Berlin and New York.

Marki, L. & Szabados, J., 1975. Interpolation and best polynomial approximation
in the domain of /?-adic numbers. Analysis Mathematica, Hungarian Academy
of Science.
Muller, H., 1976. Teil II. J. reine angew. Math. 286-7, 26-32.
Schikhof, W. H., 1975. The set of derivatives in a non-archimedean field. Math.
Ann. 216, 67-70.
van der Put, M., 1967. Algebres de fonctions continues /?-adiques, Thesis Utrecht.

Notation
Q the field of rational numbers 3
R the field of real numbers 3
C the field of complex numbers 3
Z the ring of rational integers 3
J the set {0, 1, 2,.. .} of non-negative rational integers 85
K a ring or field 17
w(a) a valuation or pseudo-valuation of K
{K}w the ring of fundamental sequences of K relative to w 21
P the ideal of null sequences of K relative to w 22
Kw the completion of K relative to w 23, 29
\a\ the absolute value of a rational, real, or complex number 8
\a\g the g-adic value of a rational or g-adic number 7
\a\p the p-adic value of a rational or p-adic number 7, 33, 73
(¾ the ring of g-adic numbers 33
Ig the ring of g-adic integers 39
<2p the field of p-adic numbers 33
IK the ring of p-adic integers 39
Kp = Qp(yfd) a quadratic extension field of Qp 66
lK the ring of Kp integers 75
A = <^41? A2,,,,, Ak > the decomposition of the g-adic number A into
its pK-adic components 55
d(a, b) the distance in the metric space (£, d) 86
L/(a, p) the balls in (£, d) defined by d(x, a) < p 87
U(a; s) the balls ^/(x, a) < g~s in (20, ^) where ^/(x, a) = |x — a\g and
seJ 89
/ usually a function J -> Qg or a function J -^Qp 104
F usually a continuous function Ig-^Qg or a continuous
function /p -»gp 110, 113

Index
abbreviated notation for g-adic and
/7-adic numbers 39
addition formulae for f and F 153
Ahlswede, R. 177
algebraic numbers and integers 145
analytic functions on /, Ip, and
IK 233, 303
balls U(m;M) 249,253
basic solutions of difference
equations 278
Bernoulli, J. 285
Bernoulli function 297
Bernoulli numbers 286
Bernoulli polynomials 286
Bojanic, R. 152, 177
bounded functions 96, 110
Cassels, J. W. S. 201,236
Clausen, Th. 291
closed sets 87
closure of a set 87
coefficients an of for F 122, 130
compact sets 88, 91, 92, 93
complement of a set 87
conditions for continuity 96, 105, 110
conditions for convergence 33
conditions for differentiability 194
conditions for the existence of the
definite integral 278, 279, 280
conditions for strict
differentiability 207
connected sets 88, 89
continuity of a function 96
decomposable functions on Ig 154
definite integral 275
derivatives 188, 220
Dieudonne?, J. 138, 197, 234
Dieudonne? (indefinite) integral 234
difference equations 276, 278
differentiability 188
digits to the bases g or p 5
elementary /?-adic functions 226
exponential series 231
frontier points 87
frontier of a set 87
function ax 226
functions of two variables 180
functions on a compact set 97
functions which are continuous, but
not differentiate 197, 199
generating series 126
higher derivatives 220
interior points 87
interior of a set 87
interpolation coefficients and
series 122, 124, 130
limits in a ring with a pseudo-valuation
(valuation) 25
locally constant functions 116
metric spaces 85
Morita, Y. 297
Miiller, H. 164
(N), the property 129
N6rlund,N.E. 281
normed solution of difference
equation 282
open sets 87
ordered rings 95
partitions of a set 88
property (N) 129
property (W) 131
pseudo-constants 210
pseudo-valuations 17
representations to the basis g or
P 11,38

320
restriction to J of a function on Ig or
IP U3
roots of unity 147
rules for differentiation 217
step functions 117, 118, 146
strictly differentiable functions 206
Tauberian lemma 191
uniform continuity of a function 97
uniform convergence of a series 100
uniform limits 100
Index
valuations 17
van der Put, M. 252, 254, 264
van der Put (indefinite) integral 264
van der Put series 254, 263
von Staudt, K. G. Chr. 291
(W), the property 131
Waldschmidt, M. 164, 165, 171
Weisman, C.S. 205, 207, 213, 216
w{t) 173