The residue classes a modp" E Z/p'tZ are expressible

ARGAND

Theorem 1. The residue classes a modp" E Z/p'tZ are expressible

uniquely in the form

a

ao +alp+02p2 + ...

+ ^modp'2

where 0a,

^<pfor

i=0,...,n—

^1.

Proof (by induction). The complete theorem is obviously true for n = 1.

If we assume the statement to be true for n — ^1, then we have an unique representation

a =

^a0+ a1p + a2p2 + .^.. +an...2p" ²+

g with 0 <

<p then

is

uniquely defined, and the asserted congruence therefore holds.

0

Every

integer f, and more generally every rational number I =

g/h whose denominator h is not divisible by p, now defines a sequence of residue classes

=

f

mod p" E n = 1,2,...

and by the theorem above, we have

10 =Go modp

Ii =

^{ao + a1pmodp2}

= ^a0+aip+a2mod?

with uniquely defined and unchanging coefficients

ao,ai,a2,...E{0,1,...,p—1}.

The number sequence

n=1,2,...

defines a p-adic integer

Zp.

We call this the p-adic expansion (or p-adic representation) of f. If, more generally, f E Q is an arbitrary rational number, we write

with

(gh,p)=1,

and if

is the p-adic expansion of g/h, then we assign to f the p-adic ^number +

a1pm41+

+ am + t2m+IP+ Qp

as its p-adic expansion.

In this way we obtain a canonical mapping

which maps Z into Zr,, and which by virtue of the uniquenes8 statement ifl Theorem 1, is injective. We now identify Q with its image in

Q

ç Q,, and Z c

Z,,, thus obtaining for every rational number f E Q an equation

f

=

and thereby establishing the analogue which we sought of the power series expansion in the theory of functions.

Ezamples.a) —1=(p—1)+(p—1)p+(p—1)p2+...

We have

—1 =(p— l)+(p— l)p+•••+(p— _—p's, also

—1 (p— 1)+ (p _— l)p+ .. + (p — mod p'.

b)

We have also

Addition and multiplication can be defined for p-adic numbers, whereby

7h,, becomes a ring whose quotient field is A straightforward attempt

to detIne the sum and product by adopting the usual "carry" rules to

which we are accustomed in ordinary decimal operations leads, however, to some significant complications. These disappear if we make use of a slightly different representation of the p-adic numbers I = avp" in which we regard them, not as a sequence of the integer partial sums

=

avpv Z, but as a sequence of the residue classes

= s, mod^pfl

The terms of this sequence lie in different rings but they are all related to one another through the canonical projections

7L/p7L Z/p2Z Z/p3Z

and the relation =

i,

holds.

We now consider, in the direct product

= {(xfl)flEu I Zn E Z/,?Z}

all those elements (xfl)flEs havingthe property forall

n=1,2

This set is called the inverse limit of the ring Z/p"Z and is denoted by

= E fl

⁼

n = 1,2,.. .}.

The modified representation of the p-adic numbers to which we referred earlier is now obtained through the following.

Theorem 2. If we associate with each p-adic integer

I =

₌

the sequence (in)nEM of residue classes i,, = mod p' E we obtain a bijection

-c-.

0

The proof is an immediate consequence of Theorem 1. The projective limit limZ/p"Z now has the advantage of being a ring, in a direct fashion, namely a subring of the direct product Z/p'Z, in which addition and multiplication are defined componentwise. If we identifS Z,, with limZ/p"Z, then Z,, also becomes a ring, the ring of p-adic integers.

As every element I Q,, has a representation

f=p-.mg

with g E addition and multiplication in Z,, can be extended to and 0,, becomes the quotient field of Z,,.

In Z,, we were able to rediscover the rational integers a Z in the guise of those p-adic numbers whose expansions ao+alp+a2p2+ .• ^werederived from the congruences

a + a1p + ...+

0 a1 <p. Through the identification

= Jim Z/p'Z

Z therefore goes over into the set of tuples

(amodp,amodp2,amodp3,...)

and thus becomes a subring of Similarly Q ^becomes a subfield of Q,,, thefield of p-adic numbers.

In we shall give a new definition of the p-adic numbers closely imitating that of the real numbers, which will bring out in an entirely straightforward way the ring and field structure of and

Corresponding to the familiar results on the decimal representation of rational numbers, we have, for p-adic numbers the following expansion the- orem.

Theorem 3. A p-adic number a

₌ avp" E _Q,, ^is rational if and

only ^if the sequence of digits (as) ^is periodic from some point onwards (that is, a finite number of digits before ^the beginning of the first period is allowed).

Proof.We may obviously assume that m = 0and a0 0. Let the sequence of digits (as) be periodic, that is to say, of the form

(a0, Oi, .. .) = ^(b0,b1, ..^{. c0,}

cj, ...

, ce_i),

where the line above the letters c indicates the principal period. We write

— n—i

,

sothat

a be rational. To prove the periodicity of the p-adic rep- resentation of a, it suffices to bring a into the above form, namely

a — — 1) —

c being integers such that

0b<ph, 0<c<p".

For we then have

0 ^b1

and the substitution of these p-adic representations gives us, by the argu- ment above, the (non-periodic) pre-period b0,^{b1,.. . ,} and the principal period

Since m = 0,the denominator f of a is prime top and thus pft 1 mod f for a suitable n. We can therefore write

apower such that

or

_ph<g<0

depending on whether a 0 or a < 0. Since (p's — i,ph) _{= 1, we can put} 9 = b(? ^{— 1) —}

with 6, c Z, and at the same time prescribe that c shall belong to any arbitrarily specified system of representatives mod(p" — 1). We stipulate

0<c<p"—2 or

depending on whether a 0 or a < 0. In both cases 0 <c < p's, as required, and it follows from

b(p"—

in both cases that 0 ( 6 <p" as required.

0

§2. THE ARITHMETIC SIGNIFICANCE OF THE p-ADIC NUMBERS

Despite being colored by their function theoretic origin, the p-adic numbers fulfill their true destiny in the realms of arithmetic, and indeed in one of its classical heartlands, the theory of Diophantine equations. A Diophan^tine problem is one in which we are given an equation

F

is a given polynomial in one or more variables, z1, ... ,x,, and are asked for its solutions in integers. This difficult problem can be attacked by weakening the question and considering instead the set of congruences for all m:

F(zi,... ,x,,) = ⁰mod m

or, what amounts to the same thing because of the Chinese Remainder Theorem, the set of congruences

for all prime powers. We might hope that from the exi8tence or nonexistence of 9olutions to the congruences, we might be able to draw corresponding conclusions about the original equation. For a fixed prime p, the infinite set of these congruences can now, with the help of the p-adic numbers, again

be expressed as a single equation. This comes from the following.

Theorem 4. Let F(xi,..

,x,) be a polynomial whose coefficients are ra- tional integers and p a fixed prime. The con gruence

is solvable for arbitrary i' 1 if and only if the equation

is solvable in p-adic integers.

Proof. We interpret the ring as in §1, as the inverse limit

= limZ/p"7Z C

The equation F =

0 factorizes, in the ring on the right, into components over the individual rings and thus into the congruences

If now

, (v)

— ,. .. )vEN E

(z1(v))vEN Z1, = limZ/p Z, is a p-adic solution of F(xj,.. ^., = 0,then the congruences are solved by

&'=1,2

Conversely, let us suppose that for every ii I we are given a solution

(zr,.

^{.. ,} of the congruence

If the elements E Z/pt17L aleady lay in limZ/p"Z for all

i =

^{1,.. ., n,} then we would have a p-adic solution of the equation F =

0. Since this is not automatically the case, we shall form the sequence

.. , a subsequence that meets our wishes. To keep the notation simple we shall deal only with the case of one variable (n = 1) and write for The general case can be proved in exactly the same way. As Z/pZ

is finite, there are infinitely many terms of which are congruent modulo p to a fixed element E Z/pZ. We can therefore choose a subsequence

of with

yj modp and

0 mod p.

In the same way we can select from {$1)} asubsequence with

mod p2 and Omodp2,

where 112 E Z/p2Z since obviously _{112 =} 111mod p. If we continue in this way we obtain, for every k 1, a subsequence of whose terms satisfy the congruences

and F(4k))EOmodpk

with certain 11k E Z/pkZ, for which mod

The 11k thus define a p-adic number y = ^E limZ/pkZ _{Z,, such}

that

F(yk) ^Omodpk

forallk?1,thatis,F(y)=0. 0

Example. Consider for v 1 the congruences 2 mod 7".

For v = ¹ the congruence has the solutions (1)

Now let a' = ^2.CLearly (2)

x2 = mody, and thus a solution of (2) must be of the form + 7t,

so that a solution of (2) must be of the form +3 + 7t. If we substitute

= ³+ 7t1 in (2), we obtain

(3+7t1)2 14-6i1

lmod7

and we thus get as a solution to z2 2 mod 72,

3+1 •7mod72.

For v = 3, we find z2 = + 72f2, ^the value t2 2mod73, and thus, for the congruence

2 mod⁷³ the solution

3+1

It is easily seen that this process can be continued indefinitely, so that one obtains a 7-adic solution

of the equation x2 = ^2. This is denoted by but is nevertheless to be strictly distinguished from the square root of 2 lying in the field JR.

If the polynomial F(xt, .. ^,z,,) is homogeneous and of degree d 1, then the equation F =^0,obviousLy always has the trivial solution (0,.. .^,0), and the question of interest is whether it has any nontrivial solutions, and if so what they are. The proof of Theorem 4 can now be modified slightly to show that the congruences

pTM

have a nontrivial solution for all ii 1 if and only if the equation

has a nontrivial p-adic solution.

At the beginning of this section we mentioned the question of whether from the solvability of an equation F = 0 in Z,, for all primes p (that is, the existence of a common solution to all the congruences F 0 mod m) one can deduce the solvability of F = 0 in rational integers. This deduction can very seldom be made (that is, the condition mentioned, though obviously necessary is rarely sufficient). However, in the case of quadratic forms we have the following so-called "local-global principle" of Minkowski—Hasse, which we state here without giving any proof (see [lJ, §7).

Theorem 5. Let F(zi,.

.. ^,z,,) be a quadratic form with rational coeffi- cients. The equation

has a nontrivial solution in Q,

if

and only if it has a nontrivzal solution in JR and in for all primes p.

THE ANALYTICAL NATURE OF p-ADIC NUMBERS The series representation

(1)

ap-adic integer bears a close ainularity to the representation of a real number between 0 and 10 as a decimal, that is, as the sum of a series of decimal fractions

ao+ai +02(1)2 0< aj <10.

However, unlike the latter, the p-adic series does not converge. Despite this nonconvergence however, the field Q, of the p-adic numbers can be constructed from the field Q in virtually the same way as the field R is constructed from Q. This is done by replacing the usual absolute value in Qbya new "p-adic" absolute value whichhas the effect of making the series (1) converge, and which enables the p-adic numbers to be regarded, in the usual way, as limits of Cauchy sequences of rational numbers.

The p-adic absolute value is defined as follows. Let a = E Q ^be a nonzero rational number with b, c E Z and _{E Q.} We divide 6 and c by

the prime p as many times as is possible, so that

(2)

(b'c",p)=1,

and define

1

lal

=—.

ptm

The p-adic value is thus no longer a measure of the absolute magnitude of a number a N, but rather has the property of being small when a is divisible by a high power of p. In particular the partial sums associated with a p-adic series 00 +alp + +... form a convergent sequence with respect to the valuation

The exponent m in the representation (2) of the number a is denoted

by and one writes formally v,(0) = oo. We have thus obtained a function

—. ZU{oo}

with the following three properties, which are easily verified:

1)

2) = v9(a)

+

3)

+ b)

where oo is a symbol 8atisfying the relations x + 00 = oo and oo > x for allx Z. The function is called the p-adic exponential valuation of Q.

Thep-adic absolute value is given by

I Q ^{R, a '—. laip}

and, in viewof the relations 1), 2), 3) it satisfies the conditions for a norm on Q, namely:

1)

Ia+ bI,, maxflalp, +

It ^{can be} shown that with _(,, and I I we have essentially exhausted the norms which can exist on Q, in

the sense that any other norm is a

power or I' ofone of these, where 8 ^is a positive real number. The ordinary absolute value (is, for good reasons which we shall not go into here, denoted by ( Along with the absolute values I

it

satisfies the following important closure relation.

Theorem 6.

For every nonzero rational integer a

HIalp=1

where p runs through all the primes, including the so-called infinite prime.

Proof. In the canonical factorization of a

a—± Hp"

the exponent of p is simply the exponential valuation and the sign is equal to Theequation can therefore be written in the form

so that in fact = ^1.

0

We shall now redefine the field Q,,ofp-adic numbers, following the same procedure as in the construction of the field of real numbers. We shall then go on to show that this new analytical definition is completely equivalent to the Hensel definition which was motivated by ideas from the theory of functions.

By a Cauchy sequence, with respect to we mean a sequence of rational numbers such that, to every s > 0 there corresponds a natural number flo for which

Ixn —

<c for all n,m> n0.

Example. Any formal series

>avp",

provides, through its partial sums

=

an example of a Cauchy sequence, since for all n > m,

—XmIp

=

A sequence {x,) in Q is called a null sequence w.r.t. _',,,

if

^is a sequence converging to zero in the usual sense.

Example. 1,p,p21p3

The Cauchy sequences form a ring R; the null sequences a maximal ideal m in R. We define the field of p-adic numbers as the residue class field

R/m.

We can embed Q in Q,,, by assigning to every a E Q the residue class rep- resented by the constant sequence (a, a, a,. _..). The p-adic absolute value

can be extended from Q to by defining, for any element z =

mod in Rim the value

urn E R.

n—.00

The limit exists, because isa Cauchy sequence in R, and it is indepen- dent of the choice of the sequence in its residue class mod m, because for any p-adic null sequence {yn) E in the relation _lYnjp 0 cer- tamly holds.

The exponential valuation Vp of Q can also be extended to the exponential valuation

-.

by defining Vp(Z) = —Log,,

or what amounts to the same thing, by

defining Vp(Z) = where x is the class of rational Cauchy sequences We again have

=

p_vF(r)

Since the image of QS under the mapping is the discrete set Z, the same is true of the image of Q, that is, v9 is a surjective homomorphism

As with the real numbers, it can be proved that the field Q,, ^is complete with respect to the p-adic absolute value _i,,, thatis, every Cauchy sequence in isconvergent with respect _i,,.Accordingly, for each prime number

p, we can associate, a

the field of fields

Q2, Qs, =IR.

An important peculiarity of the p-adic valuation _I,, is

that it not only

satisfies the usual triangle inequality, but also the stronger inequality

Jz+ ^yip <max{IxJ,,, this can be deduced a remarkable result.