The set

ARGAND

Theorem 7. The set

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.

Proof. If

E and Vp(Z) = ⁱⁿ

Z, then vp(xpm) =

^0, and hence

= ^1, so that u =

z;.

^Thisproves (i),^while (ii) is a special case of (iii). ^Suppose h 0, Z,, to be an idealof ^Let x^{= ptmu, 7L;,} be an element of ?i with the smallest m (since IxIp < 1 in must be greater

than 0). Then h =

because if y =

h, ii' E z;,

then n m

and thus y = E pmZ,,.

0

We now consider the homomorphism

Z —

Z,,

^{a '—'a}mod

whose kernel is p"Z. This homomorphism is surjective. To prove this one can easily see that the numbers z E are alrea4y limits of rational inte- gers, and ifa E Z with

Iz-aI

then ^— a) =

m

n, that is, z — a = pmu E and ^{so a =}

The homomorphism is therefore in fact an isomorphism .Z

In §1, we defined p-adic integers as formal^series

>awp",

and identified them with the sequences - _—

— , n — A,

where runs through the partial sums defined by

=

These sequences define the inverse limit

=

{(Xfl)flEii E

JJZ/I?'Z i

^xn÷i

xn}

and we looked upon the p-a.dic integers as the elements of this ring. Since

we obtain, for each n 1, a surjective homomorphism

and it is clear that the family of these homomorphisms gives us a homo- morphism

—' limZ/p'Z.

The identification of the new analytical definition of Z1, (and thus of Q,,) with the older Hensel definition can now be made.

Theorem 9. The homomorphism

—' hmZ/p"Z

is an isomorphism.

Proof. If z

is mapped onto zero, this means that z E for all n, that is, for all n, and hence = ⁰

so that z =

^0. This proves injectivity.

An element limZ/p"Z is given by a sequence of partial sums

a,, =

0 <p".

We saw earlier that this sequence is a Cauchy sequence in Z,, and thus converges to an element

z =

E

4

Since

x — =

> a,,p'

it follows that z s, for all n, that is, that x is mapped onto the element of limZ/p"Z corresponding to the sequence (sn)nEN defined above.

This proves surjectivity.

0

We emphasize that the elements of the right-hand side of -*

are given formally by the sequence of partial sums

= a,,p", n = 1,2

On the left-hand side, however, these sequences considered with respect to their absolute values converge and represent in the familiar fashion the elements of 74, as convergent infinite series

x =

The isomorphism 74, limZ/p"Z gives us additional information about the topology on Q1,, defined by the absolute value j,. The direct product

Z/i?'Z

in fact, has the product topology, in which the individual factors are re- garded as topological spaces endowed with the discrete topology. Since these factors are compact, the product is compact as well (by Tychonoff's theorem).

It can now easily be shown that the inverse limit is a closed subset of this product, and is likewise a compact space. It is also not difficult to verify that the ring isomorphism

74, — limZ/p"Z

is also a homeomorphism between topological spaces. Consequently, 74, is

a compactum, and since

also an open subset of Q,,. Every element a E therefore possesses, in a + 74,, ^an open compact neighborhood. We have therefore proved the

following.

Theorem 10. The field is locally compact. 0

The considerations in this section appear to release the p-adic numbers from their original role, modeled on that of the analytic functions, and to bring them into a closer analogy with the complex numbers themselves. It is particularly remarkable that in recent time a p-adic theory of analytic functions has been developed, in which p-adic numbers have replaced com- plex numbers both as arguments of the functions and as functional values.

This theory was initiated by the American mathematician .1. Tate, and has been widely developed by the two German mathematicians H. Grauert and ft. Remmert.

§4. THE p-ADIC NUMBERS

A far-reaching theory can be built up on the basis provided by the p-adic numbers, namely, the theory of algebraic extensions of the field 0,,, or, to express it in another way, the theory of algebraic equations

f(x)

= + + + a0 = ⁰

in one variable. We saw, in Section 2, that the solvability of such an equa-

tion in the ring Z,,,

is equivalent to the solvability of the congruences

f(x)

0 mod pP for all ii. Of fundamental importance here is the fact that a sufficient condition for this is that the congruence

f(z)E Omodp

should be solvable, as long as one restricts oneself to simple zeros. More generally, we have the important result.

Hensel's Lemma. If a polynomial f(z)

Z,,[x] has the decomposition modulo p

1(x) go(x)ho(x)modp

where the polynomials go,ho Z1,[xl ^are coprime modulo p, and ifgo is monic, then there exists a decomposition over

f(x) =

g(x)h(x)

with polynomials g, h E Zp[x), such that g(r) is monic and g(z) 90(z)mod p, 4(z) ho(x) mod p.

Proof. Let d =

deg(f),

ni =

deg(go) and without. loss of generality let us suppose that deg(ho) <d—m. We then put the polynomials g and 4, which have to be determined, into the form

g = go+ YIP+Y2P2 + h = h0 + z1p+ 22P2 +"

with polynomials y•, z Z,,[x] of degrees <m and <d — in respectively.

We now determine the polynomials

9n90+Y1P+"+.Yn_1P

n—i

L ....L n—I

— "0+ Zjp+ •+ Zn-.IP successively, in such a way that

(*)

holds for each n in turn. The equation f = gh will then hold by _{a passage}

to the limit. For n = 1, the congruence (*) is the hypothesis stated in

the lemma, and we assume that its truth has been established for n. The requirement for g,,+i, in view of

— fl 8.

_I.

⁰

— + YnP , "n+i — "n +^Z,7) then becomes

or, after division by p"

9nZn +

+

1, modp,

where = ^— Since go and h0 are coprime in ]F,,[x), there must be polynomials y, E Zp[xJ of the required kind, and can be chosen to be reduced to its minimum residue modulo go, so that

d in and deg(f0) d, it follows that deg(goz0) <d and hence d — in as required.

0

Example. The polynomial ^{— 1} splits into separate linear factors in the residue class field = By (repeated) application of Hensel's lemma, therefore, it also snlits into linear factors in

(that is, linear

factors whose coefficients belong to Q,,) and we obtain the surprising result that Q, contains the (p — 1)th roots of unity.

We now consider the finite algebraic extensions of Q,,.In contrast to the field R, the field possesses many such extensions. However, just as in the case CuR the topological structure of the ground field is extended on each extension field. More precisely, we have the following.

Theorem 11.

^Let be a finite extension of degree n. Then the absolute value I of Q,, can be extended to an absolute value _'p on K, namely, by defining

=

where N denote8 the norm of K is likewise complete with respect to

Proof. The properties alp

= 0 a = 0 and

1a131p= Iafp

clearly hold, the latter because of the multiplicativity of the norm. We shall prove the stronger version of the triangle inequality

Ia + max{IaIp,IflIp}

with the help of Hensel's Lemma. After dividing by a or /3, this reduces to checking that

IaIp <1 Ia— ^1.

In view of the transitivity of the norm we may assume for this purpose that

K = If

therefore

1(x) = +

+••• + a, then N(a) =

a

—1, that is, N(a — 1) = ±(1 +aj+.^.

We have to show therefore that Ii + +

+

1 if Ianlp 1, or in other words

We shall in fact show that E Z,, implies that as,. ..^, E Assume for the sake of argument that .. .^, were not all in Z,,.

We then multiply 1(z) by ptm, the smallest positive power of p required to ensure that all the coefficients b of

fo(z) = ptm 1(x) = 60x" + +

+ +

lie in 74,. These coefficients have the property that Omodp, while ba,.. .

are not all congruent to zero mod p. Among those coefficients is therefore a last nonzero coefficient satisfying 0 mod p. This means that there is now a factorization

fo(x)

(bog + + ..

+ modp

into factors which are relatively prime modulo p. By Hensel's lemma it follows that 1(x) is reducible, which contradicts the definition of 1(x) thus disprovLng the assumption.

The completeness of K is established by the familiar arguments, just as with R-vector spaces, by choosing a basis w1,.. . of and showing that a sequence

=

a Cauchy sequence in K if and only if the coefficient sequences lai,},..., {anj} are Cauchy sequences in

To prove the uniqueness of the extension let II beany arbitrary extension of I ', on K. Then, for

< ^{1 —}

lirn&'

= 0.

If we write, in terms of the basis w1,

...

of K,

=

+ .•.

+

then this is equivalent to the statement that = 0 in forall I = 1,. .., n. The inequality IaI < 1 thus does not depend on the choice of the extension. In other words if I Ii and I are any two extefl8iOns

(s) _{lou < 1} < 1.

Suppose now that a fixed element with 0 < looli < 1. For an arbitrary a 0 we now consider all k,l Z, I ⁰such that

< la'lj,

or, in other words, such that

log bit

I

By virtue of(s) the fractions are at the same time all rational numbers satisfying

log10012

It follows from this that log lOlt = log 1012

that is ^{Jog lou}

logbaola log ' log1012 log 10012

so that font = As I It and 112 coincide on ^S must be equal to 1, and hence Iii ₌ 112. This completes the proof of Theorem 11.

0

The field C),, of the p-adic numbers passes on many of its properties to its finite extension K. The subset

o

= ^{or _{K I} lam,, 111

again forms, just like 74, a ring with the group of units

0' = (a E K I

lamp 1}

and the single maximal ideal

p = (or _{K I} 1a1p < 1).

The residue class field = 0/p is a finite extension of the residue class field ic(p) = Z,,/pZ,, = IF,, and consequently, a finite field W,. For these reasons K is known as a p-adic number field and its elements are known as p-adic numbers.

As 0,, is locally compact, it is clear from the basis representation K =

Q,,w1 +

...

+ that every finite extension K of is likewise locally

compact. Conversely, it can be shown that the finite extensions of the fields .. ., ^{= JR}constitute precisely the totality of all the nondis- crete locally compact topological fields of characteristic zero (see [8], Ch. I

§3, Th. 9).

An important objective of the theory of numbers is that of obtaining an overall view of the finite extensions of the field One of the most beautiful and profound theorems gives a complete answer to this question, as long as we confine ourselves to Abelian extensions, that is, to finite Galois extensions whose Galois group is commutative. In these circumstances we can take as ground field an arbitrary p-adic number field K instead of Q.

If LIK is a finite extension, then we may take NLIK(L') ç K

to be its

norm group.

Theorem. Let K be a p-adic number field. The mapping LIK u—' NLIK(L)

is a one-to-one correspondence between the Abelian extension L of K and the subgroups I of K of finite index. With this relationship we even have, for Galois group G(LJK), a canonical isomorphism