[Grundlehren der mathematischen Wissenschaften] Algebraic Number Theory Volume 322 || The Theory of...

Chapter II

The Theory of Valuations

§ 1. The p-adic Numbers

The p-adic numbers were invented at the beginning of the twentieth century by the mathematician KURT HENSEL (1861-1941) with a view to introduce into number theory the powerful method of power series expansion which plays such a predominant role in function theory. The idea originated from the observation made in the last chapter that the numbers fEZ may be viewed in analogy with the polynomials fez) E C[z] as functions on the space X of prime numbers in Z, associating to them their "value" at the point p EX, i.e., the element

f(p) := f mod p

in the residue class field K (p) = Z / pZ. This point of view suggests the further question: whether not only the

"value" of the integer fEZ at p, but also the higher derivatives of f can be reasonably defined. In the case of the polynomials fez) E C[z], the higher derivatives at the point z = a are given by the coefficients of the expansion

fez) = ao + al (z - a) + ... + an(z - a)n,

and more generally, for rational functions fez) = g(z) E C(z), with h(z)

g, h E C[z], they are defined by the Taylor expansion

00

fez) = L av(z - a)V, v=o

provided there is no pole at z = a, i.e., as long as (z - a) f h(z). The fact that such an expansion can also be written down, relative to a prime number p in Z, for any rational number f E Ql as long as it lies in the local ring

Z(p) = { fig, h E Z, p f h},

leads us to the notion of p-adic number. First, every positive integer fEN admits a p-adic expansion

J. Neukirch, Algebraic Number Theory© Springer-Verlag Berlin Heidelberg 1999

100 Chapter II. The Theory of Valuations

with coefficients ai in {O, 1, ... , P - I}, i.e., in a fixed system of representatives of the "field of values" K(p) = IF p. This representation is clearly unique. It is computed explicitly by successively dividing by p, forming the following system of equations:

1= ao + ph, Il=al+p!2,

In-l = an-l + Pin,

In = an· Here ai E to, 1, ... , p - I} denotes the representative of Ji mod p E Zj pZ. In concrete cases, one sometimes writes the number I simply as the sequence of digits ao, a 1 a2 ... an, for instance

216 = 0,0011011

216 = 0,0022

216 = 1,331

(2-adic),

(3-adic),

(5-adic) .

As soon as one tries to write down such p -adic expansions also for negative integers, let alone for fractions, one is forced to allow infinite series

00

L avpv = ao +alP + a2p2 + .... v=o

This notation should at first be understood in a purely formal sense, i.e., L~o avpv simply stands for the sequence of partial sums

n-l

Sn = L avpv , n = 1,2, ... v=o

(1.1) Definition. Fix a prime number p. A p-adic integer is a formal infinite series

2 ao +alP +a2P + ... , where 0 ::: ai < p, for all i = 0,1,2, ... The set of all p-adic integers is denoted by Z p.

The p-adic expansion of an arbitrary number I E Z(p) results from the following proposition about the residue classes in Z j pn Z.

(1.2) Proposition. The residue classes a mod pn E Z j pn Z can be uniquely represented in the form

a == ao + alP + a2p2 + ... + an_lpn-l mod pn

where 0::: ai < p for i = 0, ... , n - 1.

§ 1. The p-adic Numbers 101

Proof (induction on n): This is clear for n = 1. Assume the statement is proved for n - 1. Then we have a unique representation

a = ao + alP + a2p2 + ... + an_2pn- 2 + gpn-I,

for some integer g. If g == an-I mod p such that ° ::::: an_1 < p, then an-I is uniquely determined by a, and the congruence of the proposition holds.

D

Every integer J and, more generally, every rational number J E Z(p) the denominator of which is not divisible by p, defines a sequence of residue classes

5n =JmodpnEZjpnZ, n=I,2, ... ,

for which we find, by the preceding proposition,

51 = ao mod p,

52 = ao + alP mod p2,

53 = ao + alP + a2p2 mod p3, etc.,

with uniquely determined coefficients ao, aI, a2, ... E {a, 1, ... , p-l} which keep their meaning from one line to the next. The sequence of numbers

2 n-I 1 2 Sn=aO+aIP+a2P + .. ·+an-IP , n= , , ... ,

defines a p-adic integer

We call it the p-adic expansion of J.

In analogy with the Laurent series J (z) = L~=-m av (z - a) v, we now extend the domain of p-adic integers into that of the formal series

00 "V -m -I L. avp =a_mP +···+a_IP +aO+aIP+"', v=-m

where m E Z and ° ::::: av < p. Such series we call simply p-adic numbers and we write Qp for the set of all these p-adic numbers. If J E Q is any rational number, then we write

J g -m = - p h

where g, h E Z , (gh,p) = 1,

and if


is the p-adic expansion of f' then we attach to f the p-adic number

-m -m+l t1l\ aop +alP + ... +am +am+IP + ... E'V!.p

as its p-adic expansion. In this way we obtain a canonical mapping

which takes Z into Zp and is injective. For if a, b E Z have the same p-adic expansion, then a - b is divisible by pn for every n, and hence a = b. We now identify <Q with its image in <Qp' so that we may write <Q ~ <Qp and Z ~ Z p' Thus, for every rational number f E <Q, we obtain an identity

00

f = L avpv. v=-m

This establishes the arithmetic analogue of the function-theoretic power series expansion for which we were looking.

Examples: a) -1 = (p - 1) + (p - l)p + (p - l)p2 + .... In fact, we have

-1 = (p - 1) + (p - 1) p + ... + (p - 1) pn-l _ pn,

hence -1 == (p -1) + (p - l)p + ... + (p - l)pn-1 mod pn.

b) _1_ = 1 + p + p2 + .... I-p

In fact,

1 = (1 + p + ... + pn-I)(I_ p) + pn,

hence 1 n-l n -- == 1 + p + ... + p mod p . 1- p

One can define addition and multiplication of p -adic numbers which tum Zp into a ring, and <Qp into its field of fractions. However, the direct approach, defining sum and product via the usual carry-over rules for digits, as one does it when dealing with real numbers as decimal fractions, leads into complications. They disappear once we use another representation of the p -adic numbers f = L~=o av p v, viewing them not as sequences of sums of integers

n-l Sn = L avpv E Z,

v=o


but rather as sequences of residue classes

The terms of such a sequence lie in different rings Z I pn Z, but these are related by the canonical projections

and we find

In the direct product

00 n ZlpnZ = {(Xn)nEN I Xn E ZlpnZ} , n=l

we now consider all elements (Xn)nEN with the property that

An(Xn+l) = Xn for all n = 1,2, ...

This set is called the projective limit of the rings Z I pn Z and is denoted by ~ Zlpnz. In other words, we have

n

00

~ ZI pnZ = { (Xn)nEN E n ZI pnZ I An(Xn+l) = Xn, n = 1,2, ... } . n n=l

The modified representation of the p -adic numbers alluded to above now follows from the

(1.3) Proposition. Associating to every p -adic integer

the sequence (s n)nEN of residue classes

n-l sn = L avpv mod pn E 7lIpn71,

v=o

yields a bijection


The proof is an immediate consequence of proposition (1.2). The projective limit !!!!! Z j pn Z offers the advantage of being clearly a ring. In fact, it is a subring of the direct product TI~J Zj pnZ where addition and multiplication are defined componentwise. We identify Z p with !!!!! Z j pn Z and obtain the ring of p-adic integers Zp,

Since every element f E Qp admits a representation

f = p-m g

with g E ZP' addition and multiplication extend from Zp to Qp and Q p becomes the field of fractions of Zp.

In Z p' we found the rational integers a E Z which were determined by the congruences

a == ao + aJP + ... + an_Jpn-J mod pn,

0:::: ai < p. Making the identification

the subset Z is taken to the set of tuples

00

(a mod p, a mod p2, a mod p3, ... ) E TI ZjpnZ n=l

and thereby is realized as a subring of Z p' We obtain Q as a subfield of the field Q p of p -adic numbers in the same way.

Despite their origin in function-theoretic ideas, the p-adic numbers live up to their destiny entirely within arithmetic, more precisely at its classical heart, the Diophantine equations. Such an equation

F(xJ, ... ,xn)=O

is given by a polynomial F E Z[xJ, ... , xn], and the question is whether it admits solutions in integers. This difficult problem can be weakened by considering, instead of the equation, all the congruences

F(xJ, ... , xn) == 0 mod m.

By the Chinese remainder theorem, this amounts to considering the congruences

F(xJ, ... , xn) == 0 mod pV

modulo all prime powers. The hope is to obtain in this way information about the original equation. This plethora of congruences is now synthesized again into a single equation by means of the p-adic numbers. In fact, one has the


(1.4) Proposition. Let F (Xl, ... , xn) be a polynomial with integer coefficients, and fix a prime number p. The congruence

F(XI, ... , xn) == 0 mod pV

is solvable for arbitrary v ~ 1 if and only if the equation

F(XI, ... ,Xn)=O

is solvable in p -adic integers.

Proof: As established above, we view the ring Zp as the projective limit

00

Zp = ll!!! ZjpVZ S; n ZjpvZ. v v=l

Viewed over the ring on the right, the equation F = 0 splits up into components over the individual rings Z j p v Z, namely, the congruences

F(XI, ... , xn) == 0 mod pV.

If now ( ) _ ( (v) (v)) '7ln XI,···,Xn - Xl ' ... 'Xn vENElUp'

with (X?))VEN E Zp = ll!!! ZjpVZ, is a p-adic solution of the equation v

F (Xl, ... , Xn) = 0, then the congruences are solved by

F ( (v) (v)) - 0 d v Xl , ... , Xn = mo p , v = 1,2, ...

Conversely, let a solution (x~V), ... , x~v)) of the congruence

F(XI, ... , xn) == 0 mod pV

be given for every v ~ 1. If the elements (X?))VEN E n:IZjpVZ are already in ll!!! Z j p v Z, for all i = 1, ... , n, then we have a p -adic solution of the equation F = O. But this is not automatically the case. We will therefore extract a subsequence from the sequence (x~V), ... , x~v)) which fits our needs. For simplicity of notation we only carry this out in the case

n = 1, writing Xv = xiV). The general case follows exactly the same pattern.

In what follows, we view (xv) as a sequence in Z. Since ZjpZ is finite, there are infinitely many terms Xv which mod p are congruent to the same

element YI E ZjpZ. Hence we may choose a subsequence {x~l)} of {xv} such that

x~l) == YI mod p and F(x~l)) == 0 mod p.


Likewise, we may extract from {x~l)} a subsequence {x~2)} such that

x~2) == Y2 mod p2 and F (x~2)) == 0 mod p2,

where Y2 E Z / p 2Z evidently satisfies Y2 == Yl mod p. Continuing in this way, we obtain for each k ::: 1 a subsequence {x~k)} of {x~k-l)} the terms of which satisfy the congruences

x~k) == Yk mod pk and F (x~k)) == 0 mod pk

for some Yk E Z / pkZ such that

d k-l Yk == Yk-l mo P .

The Yk define a p-adic integer Y = (YkhEN E ~ Z/pkZ = Zp satisfying k

F(Yk) == 0 mod pk

for all k ::: 1. In other words, F(y) = o. D

Exercise 1. A p-adic number a = I::-m avpv E Qp is a rational number if and only if the sequence of digits is periodic (possibly with a finite string before the first period).

i

Hint: Writepma=b+c-l P n,O:::::b <pi,O:::::C< pn. -P

Exercise 2. A p-adic integer a = ao + alP + a2p2 + ... is a unit in the ring Zp if and only if ao =1= o. Exercise 3. Show that the equation x 2 = 2 has a solution in Z7.

Exercise 4. Write the numbers ~ and - ~ as 5-adic numbers.

Exercise 5. The field Qp of p-adic numbers has no automorphisms except the identity.

Exercise 6. How is the addition, subtraction, multiplication and division of rational numbers reflected in the representation by p-adic digits?

§ 2. The p-adic Absolute Value

The representation of a p -adic integer

(1)

§ 2. The p-adic Absolute Value 107

resembles very much the decimal fraction representation

1 1 2 ao +a1(1O)+a2 (1O) + ... , O~ai<lO,

of a real number between 0 and 10. But it does not converge as the decimal fraction does. Nonetheless, the field Qp of p-adic numbers can be constructed from the field Q in the same fashion as the field of real numbers JR. The key to this is to replace the ordinary absolute value by a new "p-adic" absolute value I Ip with respect to which the series (1) converge so that the p -adic numbers appear in the usual manner as limits of Cauchy sequences of rational numbers. This approach was proposed by the Hungarian mathematician J. KURSCHAK. The p-adic absolute value I Ip is defined as follows.

Let a = ~, b, C E Z be a nonzero rational number. We extract from b and from c as high a power of the prime number p as possible,

(2)

and we put

b' a = pm - , (b' c' , p) = 1,

c'

1 lal p = -. pm

Thus the p -adic value no longer measures the size of a number a EN. Instead it becomes small if the number is divisible by a high power of p. This elaborates on the idea suggested in (1.4) that an integer has to be 0 if it is infinitely divisible by p. In particular, the summands of a p-adic series ao + alP + a2p2 + ... form a sequence converging to 0 with respect to lip.

The exponent m in the representation (2) of the number a is denoted by vp(a), and one puts formally vp(O) = 00. This gives the function

vp : Q ---+ Z U {oo},

which is easily checked to satisfy the properties

1) vp(a) = 00 {:=} a = 0,

2) vp(ab) = vp(a) + vp(b),

3) vp(a + b) :::: min{vp(a), vp(b)},

where x + 00 = 00, 00 + 00 = 00 and 00 > x, for all x E Z. The function vp is called the p-adic exponential valuation of Q. The p-adic absolute value is given by

I Ip: Q ---+ JR, a 1---+ lal p = p-vp(a).

In view of 1), 2), 3), it satisfies the conditions of a norm on Q:


1) lalp=O{=}a=O,

2) lablp = lalplbl p,

3) la + blp .:::: max{lal p, Iblp} .:::: lal p + Iblp.

One can show that the absolute values I Ip and I I essentially exhaust all norms on Ql: any further norm is a power I I~ or I IS, for some real number s > ° (see (3.7». The usual absolute value I I is denoted in this context by I 100' The good reason for this will be explained in due course. In conjunction with the absolute values lip, it satisfies the following important product formula:

(2.1) Proposition. For every rational number a f. 0, one has

nlalp = 1, p

where p varies over all prime numbers as well as the symbol 00.

Proof: In the prime factorization

a = ± n pVp

Pi=oo

of a, the exponent vp of p is precisely the exponential valuation vp(a) and

the sign equals 1:100' The equation therefore reads

a 1 a=- n-,

la I 00 Pi=oo la Ip so that one has indeed np la Ip = 1. D

The notation I 100 for the ordinary absolute value is motivated by the analogy of the field of rational numbers Ql with the rational function field k(t) over a finite field k, with which we started our considerations. Instead of Z, we have inside k(t) the polynomial ring k[t], the prime ideals p f. ° of which are given by the monic irreducible polynomials pet) E k[t]. For every such p, one defines an absolute value

I 113: k(t) ---+ lR

as follows. Let J(t) = !i~~, get), h(t) E k[t] be a nonzero rational function.

We extract from g(t) and h(t) the highest possible power of the irreducible polynomial pet),

m get) J(t) = pet) -_ -, (g fl, p) = 1,

h(t)


and put

vp(f) = m, I/lp = q;vp(f) ,

where qp = qdp , dp being the degree of the residue class field of p over k and q a fixed real number> 1. Furthermore we put vp(O) = 00 and 10lp = 0, and obtain for vp and I Ip the same conditions 1), 2), 3) as for vp and I Ip above. In the case p = (t - a) for a E k, the valuation vp (f) is clearly the order of the zero, resp. pole, of the function I = I(t) at t = a.

But for the function field k(t), there is one more exponential valuation

Voo : k(t) -+ !Z U {oo},

namely voo(f) = deg(h) - deg(g) ,

where I = * =j:. 0, g, h E k[t]. It describes the order of zero, resp. pole, of I(t) at the point at infinity 00, i.e., the order of zero, resp. pole, of the function I (1 It) at the point t = 0. It is associated to the prime ideal p = (11 t) of the ring k[11 t] ~ k(t) in the same way as the exponential valuations vp are associated to the prime ideals p of k[t]. Putting

I I 100 = q-voo(f) ,

the unique factorization in k(t) yields, as in (2.1) above, the formula

n I/lp = 1, p

where p varies over the prime ideals of k[t] as well as the symbol 00, which now denotes the point at infinity (see chap. I, § 14, p. 95).

In view of the product formula (2.1), the above consideration shows that the ordinary absolute value I I of <Q should be thought of as being associated to a virtual point at infinity. This point of view justifies the notation I 100' obeys our constant leitmotiv to study numbers as functions from a geometric perspective, and it will fulfill the expectations thus raised in an ever growing and amazing manner. The decisive difference between the absolute value I 100 of <Q and the absolute value I 100 of k(t) is, however, that the former is not derived from any exponential valuation vp attached to a prime ideal.

Having introduced the p-adic absolute value I Ip on the field <Q, let us now give a new definition of the field <Qp of p-adic numbers, imitating the construction of the field of real numbers. We will verify afterwards that this new, analytic construction does agree with Hensel's definition, which was motivated by function theory.


A Cauchy sequence with respect to I Ip is by definition a sequence {xn} of rational numbers such that for every £ > 0, there exists a positive integer no satisfying

IXn - Xm Ip < £ for all n, m ~ no.

Example: Every formal series

00

L avpv, 0::: av < p, v=o

provides a Cauchy sequence via its partial sums

because for n > m one has

n-l

Xn = L avpv, v=o

n-l 1 IXn-xmlp=1 Lavpvl ::: max{lavpVlp}<-·

v=m p m~v<n - pm

A sequence {xn} in <Q is called a nUllsequence with respect to I Ip if IXn Ip is a sequence converging to 0 in the usual sense.

Example: 1, p, p2, p3, ...

The Cauchy sequences form a ring R, the nUllsequences form a maximal ideal m, and we define afresh the field of p -adic numbers to be the residue class field

<Qp := Rim.

We embed <Q in <Qp by associating to every element a E <Q the residue class of the constant sequence (a, a, a, ... ). The p-adic absolute value I Ip on <Q is extended to <Q p by giving the element x = {xn} mod m E Rim the absolute value

This limit exists because {Ixnlp} is a Cauchy sequence in JR, and it is independent of the choice of the sequence {xn} within its class mod m because any p -adic null sequence {Yn} E m satisfies of course lim I Yn I p = o.

n-'>oo

The p-adic exponential valuation vp on <Q extends to an exponential valuation

§ 2. The p-adic Absolute Value III

In fact, if x E Qp is the class of the Cauchy sequence {xn} where Xn f. 0, then

Vp(Xn) = -logp IXn Ip

either diverges to 00 or is a Cauchy sequence in Z which eventually must become constant for large n because Z is discrete. We put

Again we find for all x E Qp that

Ixl p = p-vp(x).

As for the field of real numbers one proves the

(2.2) Proposition. The field Qp of p-adic numbers is complete with respect to the absolute value lip, i.e., every Cauchy sequence in Qp converges with respect to I I p .

As well as the field ~, we thus obtain for each prime number p a new field Qp with equal rights and standing, so that Q has given rise to the infinite family of fields

An important special property of the p-adic absolute values I Ip lies in the fact that they do not only satisfy the usual triangle inequality, but also the stronger version

This yields the following remarkable proposition, which gives us a new definition of the p-adic integers.

(2.3) Proposition. The set

is a subring of Q p. It is the closure with respect to I Ip of the ring Z in the field Qp .


Proof: That 7Lp is closed under addition and multiplication follows from

If {xn} is a Cauchy sequence in 7L and x = lim xn, then IXn Ip ::::: 1 implies n--+oo

also Ixlp ::::: 1, hence x E 7L p. Conversely, let x = lim Xn E 7L p, for a n--+oo

Cauchy sequence {xn} in Ql. We saw above that one has Ixlp = IXnl p ::::: 1 for n ::: no, i.e., Xn = ~n, with an, bn E 7L, (bn, p) = 1. Choosing for each n ::: no a solution Yn E 7L of the congruence bnYn == an mod pn yields

IXn - Yn Ip ::::: ~ and hence x = lim Yn, so that x belongs to the closure p n--+oo

of 7L. 0

The group of units of 7L p is obviously

7L; = {x E 7Lp Ilxlp = 1} .

Every element x E Ql; admits a unique representation

x = pm U with m E 7L and u E 7L; .

For if vp(x) = m E 7L, then vp(xp-m) = 0, hence Ixp-m Ip = 1, i.e., u = xp-m E 7L;. Furthermore we have the

(2.4) Proposition. The nonzero ideals of the ring 7L p are the principal ideals

pn7Lp = {x E Qlp I vp(x)::: n},

with n ::: 0, and one has

Proof: Let a #- (0) be an ideal of 7Lp and x = pmu, U E 7L;, an element of a with smallest possible m (since Ix Ip ::::: 1, one has m ::: 0). Then a = pm7Lp because Y = pnu' E a, u' E 7L;, implies n ::: m, hence Y = (pn-mu')pm E pm7Lp. The homomorphism

7L ---+ 7L p/ pn7Lp, a ~ a mod pn7Lp,

has kernel pn 7L and is surjective. Indeed, for every x E 7L p' there exists by (2.3) an a E 7L such that

1 lx-alp::::: n'

p


i.e., vp(x - a) ::: n, therefore x - a E pnzp and hence x == a mod pnzp. So we obtain an isomotphism

o

We now want to establish the link with Hensel's definition of the ring Zp and the field <Qp which was given in § 1. There we defined the p-adic integers as formal series

00

L: av p v , 0:::: av < p, v=o

which we identified with sequences

sn = Sn mod pn E ZjpnZ, n = 1,2, ... ,

where Sn was the partial sum

n-l

Sn = L: avpv. v=o

These sequences constituted the projective limit

00

~ Zj pnZ = { (Xn)nEN E IT Zj pnZ I Xn+l t-+ xn}. n n=l

We viewed the p-adic integers as elements of this ring. Since

we obtain, for every n ::: 1, a surjective homomotphism

Zp --+ Zj pnZ.

It is clear that the family of these homomotphisms yields a homomotphism

Zp --+ ~ ZjpnZ. n

It is now possible to identify both definitions given for Zp (and therefore also for Qp) via the

(2.5) Proposition. The homomorphism

Zp --+ ~ ZjpnZ n

is an isomorphism.


Proof: If x E Zp is mapped to zero, this means that x E pnzp for all n ::: 1,

i.e., Ixlp :::: ~ for all n ::: 1, so that Ixlp = 0 and thus x = O. This shows p

injectivity.

An element of ¥!!! Z j pn Z is given by a sequence of partial sums n

n-J Sn = L avpv, 0:::: av < p.

v=o

We saw above that this sequence is a Cauchy sequence in Z p' and thus converges to an element

00

x = L av p v E Z p. v=o

Since 00

x - Sn = L avpv E pnZp, v=n

one has x == Sn mod pn for all n, i.e., x is mapped to the element of ¥!!! Zjpnz which is defined by the given sequence (Sn)nEN. This shows

n surjectivity. D

We emphasize that the elements on the right hand side of the isomorphism

Zp ~ ~ ZjpnZ n

are given formally by sequences of partial sums

n-J Sn = L avpv , n = 1,2, ...

v=o

On the left, however, these sequences converge with respect to the absolute value and yield the elements of Z p in the familiar way, as convergent infinite series

00

x = L avpv. v=o

Yet another, very elegant method to introduce the p-adic numbers comes about as follows. Let Z[[X]] denote the ring of all formal power series L~o aj Xi with integer coefficients. Then one has the

(2.6) Proposition. There is a canonical isomorphism

Zp ~ Z[[X]]j(X - p).


Proof: Consider the visibly surjective homomorphism Z[[X]] -+ Zp which to every formal power series L~o avXv associates the convergent series L~o av p v. The principal ideal (X - p) clearly belongs to the kernel of this mapping. In order to show that it is the whole kernel, let f(X) = L~oavXv be a power series such that f(p) = L~oavpv = o. Since Zpl pnzp ~ ZI pnz, this means that

ao + alP + ... + an_Ipn-1 == 0 mod pn

for all n. We put, for n 2: 1,

1 n I bn- l = --(aO+aIP+···+an-IP -). pn

Then we obtain successively

ao = - pbo,

al = bo - pbl ,

a2 = bl - pb2, etc.

But this amounts to the equality

(ao + alX + a2X2 + ... ) = (X - p)(bo + blX + b2X2 + ... ), i.e., f(X) belongs to the principal ideal (X - p).

Exercise 1. Ix - Yip:::: Ilxlp - Iylp I.

o

Exercise 2. Let n be a natural number, n = ao + alP + ... + ar_lpr-l its p-adic expansion, with 0:::: ai < p, and s = aO+al + .. ·+ar-l. Show that vp(n!) = n - SI.

p-

Exercise 3. The sequence 1, /O'~, tbr, ... does not converge in Qp' for any p.

Exercise 4. Let S E 1 + pZp, and let IX = ao +alP + a2p2 + ... be a p-adic integer, and write Sn = ao + alP + ... + an_lpn-I. Show that the sequence sSn converges to a number SOl in 1 + pZp. Show furthermore that 1 + pZp is thus turned into a multiplicative Zp-module.

Exercise 5. For every a E Z, (a,p) = 1, the sequence {aPn}nEN converges in Qp.

Exercise 6. The fields Qp and Qq are not isomorphic, unless p = q.

Exercise 7. The algebraic closure of Qp has infinite degree.

Exercise 8. In the ring Zp[[X]] of formal power series L:oavXv over ZP' one has the following division with remainder. Let f, g E Zp[[X]] and let f(X) = ao + alX + ... such that plav for v = 0, ... , n - 1, but P f an. Then one may write in a unique way

g = qf +r, where q E Zp[[X]], and r E Zp[X] is a polynomial of degree:::: n - 1.


Hint: Let r be the operator r(I::o bvX V ) = I::n bvxv-n. Show that U (X) = an+an+lX + ... = r(f(X» is a unit in Zp[[X]] and write I(X) = pp(x)+xnu (X) with a polynomial P(X) of degree::: n - 1. Show that

1 00 ""( P)i q(X) = -- L(-l)'p' ro - or(g) U(X) i=O U

is a well-defined power series in Zp[[X]] such that r(qf) = reg).

Exercise 9 (p-adic Weierstrass Preparation Theorem). Every nonzero power series 00

I(X) = L avr E Zp[[X]] v=o

admits a unique representation

I(X) = pll-P(X)U(X),

where U(X) is a unit in Zp[[X]] and P(X) E Zp[X] is a monic polynomial satisfying P(X) == xn mod p.

§ 3. Valuations

The procedure we performed in the previous section with the field Q in order to obtain the p-adic numbers can be generalized to arbitrary fields using the concept of (multiplicative) valuation.

(3.1) Definition. A valuation of a field K is a function

II:K~ffi.

enjoying the properties

(i) Ixl 2: 0, and Ixl = ° {:=::} x = 0,

(ii) IxYI = Ixllyl, (iii) Ix + y I ::::; Ix I + Iy I "triangle inequality".

We tacitly exclude in the sequel the case where I I is the trivial valuation of K which satisfies Ix I = 1 for all x =J O. Defining the distance between two points x, y E K by

d(x,y) = Ix - yl

makes K into a metric space, and hence in particular a topological space.

(3.2) Definition. Two valuations of K are called equivalent if they define the same topology on K.

§ 3. Valuations 117

(3.3) Proposition. Two valuations I It and I 12 on K are equivalent if and only if there exists a real number s > 0 such that one has

for all x E K.

Proof: If I 11 = I I~, with s > 0, then I It and I 12 are obviously equivalent. For an arbitrary valuation I I on K, the inequality Ix I < 1 is tantamount to the condition that {xn }nEN converges to zero in the topology defined by I I. Therefore if I 11 and I 12 are equivalent, one has the implication

Ixlt < 1 ===> Ixl2 < 1.

Now let y E K be a fixed element satisfying Iylt > 1. Let x E K, x i= O. Then Ixlt = Iylf for some a E R Let mdni be a sequence of rational numbers (with ni > 0) which converges to a from above. Then we have

Ixlt = Iylf < IYI~i/ni, hence

so that Ixl2 :::: IYI~i/ni, and thus Ixl2 :::: IYI~. Using a sequence mdni which converges to a from below (*) tells us that Ix 12 2: I y I~. SO we have Ixl2 = IYI~. For all x E K, x i= 0, we therefore get

log Ix 11 log I y It -..::.-- - _. s log Ixl2 - log lyl2 -. ,

hence Ixlt = Ixl~. But Iylt > 1 implies lyl2 > 1, hence s > o. 0

The proof shows that the equivalence of I 11 and I 12 is also equivalent to the condition

Ixlt < 1 ===> Ixl2 < 1.

We use this for the proof of the following approximation theorem, which may be considered a variant of the Chinese remainder theorem.

(3.4) Approximation Theorem. Let I 11, ... , I I n be pairwise inequivalent valuations of the field K and let aI, ... , an E K be given elements. Then for every s > 0 there exists an x E K such that

Ix - aili < s for all i = 1, ... , n.


Proof: By the above remark, since I II and I In are inequivalent, there exists ex E K such that lex 11 < 1 and lex In :::: 1. By the same token, there exists f3 E K such that I f31 n < 1 and I f311 :::: 1. Putting y = f3/ ex, one finds I y II > 1 and I yin < 1.

We now prove by induction on n that there exists z E K such that

I z 11 > 1 and I z Ij < 1 for j = 2, ... , n.

We have just done this for n = 2. Assume we have found z E K satisfying

Izil > 1 and Izlj < 1 for j = 2, "., n - 1.

If Izln :s 1, then zm y will do, for m large. If however Izln > 1, the sequence tm = zm / (1 + zm) will converge to 1 with respect to I 11 and I In, and to 0 with respect to I 12, ... , I In-I. Hence, for m large, tmy will suffice.

The sequence zm / (1 + zm) converges to 1 with respect to I II and to 0 with respect to I 12, ... , I In. For every i we may construct in this way a Zi which is very close to I with respect to I Ii, and very close to 0 with respect to I Ij for j =j:. i. The element

then satisfies the statement of the approximation theorem. o

(3.5) Definition. The valuation I I is called nonarchimedean if In I stays bounded, for all n EN. Otherwise it is called archimedean.

(3.6) Proposition. The valuation I I is nonarchimedean if and only if it satisfies the strong triangle inequality

Ix + yl :s maxI lxi, IYI}.

Proof: If the strong triangle inequality holds, then one has

Inl = 11+·,,+11:s 1.

Conversely, let Inl :s N for all n EN. Let x, y E K and suppose Ixl :::: Iyl. Then Ixlvlyln-v :s Ixl n for v:::: 0 and one gets

n

Ix + yin :s L I (~) Ilxlvlyln-v :s N(n + 1)lxln, v=o

hence

Ix + yl :s NI/n(1 + n)I/n Ixl = NI/n(1 + n)I/n maxI Ix I, Iyl},

and thus Ix + yl :s max{lxl, Iyl} by letting n --+ 00. o


Remark: The strong triangle inequality immediately implies that

Ixl # Iyl ===} Ix + yl = maxI lxi, Iyl} . One may extend the nonarchimedean valuation I I of K to a valuation of the function field K (t) in a canonical way by setting, for a polynomial I(t) = ao +alt + ... +antn,

III = maxI laol, ... , lanl}. The triangle inequality I I + g I ::::: max {I I I, I g I} is immediate. The proof that II g I = I I II g I is the same as the proof of Gauss's lemma for polynomials over factorial rings once we replace the content of I in this lemma by the absolute value I I I·

For the field Q, we have the usual absolute value I 100 = I I, this being the archimedean valuation, and for each prime number p the nonarchimedean valuation lip. As a matter of fact:

(3.7) Proposition. Every valuation of Q is equivalent to one of the valuations I Ip or I 100·

Proof: Let II II be a nonarchimedean valuation of Q. Then II nil = 111 + ... + 111 ::::: 1, and there is a prime number p such that II p II < 1 because, if not, unique prime factorization would imply IIx II = 1 for all x E Q*. The set

a. = {a E Z Iliall < 1} is an ideal of Z satisfying pZ ~ a. # Z, and since pZ is a maximal ideal, we have a. = pZ. If now a E Z and a = bpm with p f b, so that b f/. a., then Ilbll = 1 and hence

lIall = IIplim = lal~ where s = -log lip II flog p. Consequently II II is equivalent to lip.

Now let II II be archimedean. Then one has, for every two natural numbers n,m> 1,

In fact, we may write

m = ao + aln + ... + arnr

where ai E {O, 1, ... , n - I} and nr ::::: m. Hence, observing that r ::::: logmjlogn and lIadl = 111 + ... + 111 ::::: ai 11111 ::::: n, one gets the inequality

IImll ::::: L lIadl . IInlli ::::: L lIai II . IInur ::::: (1 + ~:~:)n . IIn\llogm/logn.


Substituting here mk for m, taking k-th roots on both sides, and letting k tend to 00, one finally obtains

IImll .::: lin IIlogm/logn , or IImlll/logm.::: IInlll/logn.

Swapping m with n gives the identity (*). Putting c = IInlll/logn we have IInll = clogn , and putting c = eS yields, for every positive rational number x = alb,

Therefore II II is equivalent to the usual absolute value I I on Q. 0

Let I I be a nonarchimedean valuation of the field K. Putting

vex) = -log Ixl for x =J. 0, and v(O) = 00,

we obtain a function v: K ~ ~ U {(X)}

verifying the properties

(i) vex) = 00 {::::=} x = 0,

(ii) v(xy) = vex) + v(y),

(iii) vex + y) ::: min{v(x) , v(y)},

where we fix the following conventions regarding elements a E ~ and the symbol 00: a < 00, a + 00 = 00, 00 + 00 = 00.

A function v on K with these properties is called an exponential valuation of K. We exclude the case of the trivial function vex) = 0 for x =J. 0, v(O) = 00. Two exponential valuations VI and V2 of K are called equivalent if VI = SV2, for some real number S > O. For every exponential valuation V we obtain a valuation in the sense of (3.1) by putting

Ixl = q-v(x),

for some fixed real number q > 1. To distinguish it from v, we call I I an associated multiplicative valuation, or absolute value. Replacing v by an equivalent valuation sv (Le., replacing q by q' = qS) changes I I into the equivalent multiplicative valuation I IS. The conditions (i), (ii), (iii) immediately imply the

(3.8) Proposition. The subset

0= {x E K I vex) ::: o} = {x E K Ilxl .::: 1} is a ring with group of units

o*={xEKlv(x)=O} ={xEKllxl=l}

and the unique maximal ideal

p = {x E K I vex) > o} = {x E K Ilxl < 1} .


o is an integral domain with field of fractions K and has the property that, for every x E K, either x E 0 or x-I E o. Such a ring is called a valuation ring. Its only maximal ideal is I' = {x EO I x-I fj o}. The field 0/1' is called the residue class field of o. A valuation ring is always integrally closed. For if x E K is integral over 0, then there is an equation

xn +alxn- I + ... +an = 0

with ai E 0 and the hypothesis x rt 0, so that X-I E 0, would imply the contradiction x = -al - a2x-1 - '" - an(x-I)n-I E O.

An exponential valuation v is called discrete if it admits a smallest positive value s. In this case, one finds

v(K*) = sZ.

It is called normalized if s = 1. Dividing by s we may always pass to a normalized valuation without changing the invariants 0, 0* , p. Having done so, an element

rr E 0 such that v(rr) = 1

is a prime element, and every element x E K* admits a unique representation

x = u rrm

with m E Z and u E 0*. For if v(x) = m, then v(x rr-m) = 0, hence u = x rr -m E 0*.

(3.9) Proposition. If v is a discrete exponential valuation of K, then

0= {x E K I v(x) ~ O} is a principal ideal domain, hence a discrete valuation ring (see I, (11.3».

Suppose v is normalized. Then the nonzero ideals of 0 are given by

pn = rrno = {x E K I v(x) ~ n}, n ~ 0,

where rr is a prime element, i.e., v(rr) = 1. One has

pn /pn+1 ~ 0/1'.

Proof: Let a # 0 be an ideal of 0 and x # 0 an element in a with smallest possible value v(x) = n. Then x = u rrn, u E 0*, so that rrno ~ a. If y = 8 rrm E a is arbitrary with 8 E 0*, then m = v (y) ~ n, hence y = (8 rrm-n)rrn E rrno, so that a = rrno. The isomorphism

pn /pn+1 ~ 0/1'

results from the correspondence arrn 1--4 a mod p. o


In a discretely valued field K the chain

2 3 02P2P 2P 2···

consisting of the ideals of the valuation ring 0 forms a basis of neighbourhoods of the zero element. Indeed, if v is a normalized exponential valuation and I I = q-V (q > 1) an associated multiplicative valuation, then

As a basis of neighbourhoods of the element 1 of K*, we obtain in the same way the descending chain

0* = U(O) 2 U(I) 2 U(2) 2 ...

of subgroups

1 U(n) = 1 + pn = {x E K* I 11 - x I < -}, n > 0,

qn-l

of 0*. (Observe that 1 +pn is closed under multiplication and that, if x E u(n),

then so is X-I because 11-x-11 = Ixl-1lx -11 = II-xl < L,.) u(n) q

is called the n-th higher unit group and U(I) the group of principal units. Regarding the successive quotients of the chain of higher unit groups, we have the

(3.10) Proposition. 0* /u(n) ,..., (o/pn)* and u(n) /u(n+l) = o/p, for

n::::l.

Proof: The first isomorphism is induced by the canonical and obviously surjective homomorphism

0* ---+ (o/pn)*, u t----+ u mod pn,

the kernel of which is U(n). The second isomorphism is given, once we choose a prime element ;r, by the surjective homomorphism

U(n) = 1 + ;rno ---+ o/p, 1 + ;rna t----+ a mod p,

which has kernel u(n+l). 0

§ 4. Completions 123

Exercise 1. Show that Izl = (ZZ)I/2 = JINCilR(Z)I is the only valuation of C which extends the absolute value I I of JR.

Exercise 2. What is the relation between the Chinese remainder theorem and the approximation theorem (3.4)?

Exercise 3. Let k be a field and K = k(t) the function field in one variable. Show that the valuations vI' associated to the prime ideals p = (p(t)) of k[t], together with the degree valuation voo , are the only valuations of K, up to equivalence. What are the residue class fields?

Exercise 4. Let 0 be an arbitrary valuation ring with field of fractions K, and let r = K* /0'.' Then r becomes a totally ordered group if we define x mod 0* 2: y mod 0* to mean x/y EO.

Write r additively and show that the function

v : K ---+ r U {oo},

v(O) = 00, vex) = x mod 0* for x E K*, satisfies the conditions

1) vex) = 00 ==> x = 0,

2) v(xy) = vex) + v(y),

3) vex + y) 2: min{v(x), v(y)}.

v is called a Krull valuation.

§ 4. Completions

(4.1) Definition. A valued field (K, I I) is called complete if every Cauchy sequence {an}nEN in K converges to an element a E K, i.e.,

lim I an - a I = O. n-+oo

Here, as usual, we call {an}nEN a Cauchy sequence if for every 8 > 0 there exists N E N such that

Ian - am I < 8 for all n, m 2: N.

From any valued field (K, I I) we get a complete valued field (K, I I) by the process of completion. This completion is obtained in the same way as the field of real numbers is constructed from the field of rational numbers.

Take the ring R of all Cauchy sequences of (K, I I), consider therein the maximal ideal m of all nUllsequences with respect to I I, and define

K = Rjm.


One embeds the field K into K by sending every a E K to the class of the constant Cauchy sequence (a, a, a, ... ). The valuation I I is extended from K to K by giving the element a E K which is represented by the Cauchy sequence {an}neN the absolute value

lal = lim Ian I. n-+oo

This limit exists because I Ian I - lam II :::: Ian - am I implies that Ian I is a Cauchy sequence of real numbers. As in the case of the field of real numbers, one proves that K is complete with respect to the extended I I, and that each a E K is a limit of a sequence {an} in K. Finally one proves the uniqueness of the completion (K, I I): if (K', I I') is another complete valued field that contains (K, I I) as a dense subfield, then mapping

I I -lim an 1----+ I I' -lim an n-+oo n-+oo

gives a K -isomorphism (J : K ~ K' such that lal = l(Jal'.

The fields ~ and C are the most familiar examples of complete fields. They are complete with respect to an archimedean valuation. Amazingly enough, there are no others of this type. More precisely we have the

(4.2) Theorem (OSTROWSKI). Let K be a field which is complete with respect to an archimedean valuation 1 I. Then there is an isomorphism (J from K onto ~ or C satisfying

lal = l(Jal s forall a E K,

for some fixed S E (0, 1].

Proof: We may assume without loss of generality that ~ ~ K and that the valuation I I of K is an extension of the usual absolute value of R In fact, replacing I I by I Is- 1 for a suitable S > 0, we may assume by (3.7) that the restriction of I I to Q is equal to the usual absolute value. Then taking the closure ij in K we find that ij is complete with respect to the restriction of I I to ij, in other words, it is a completion of (Q, I I). In view of the uniqueness of completions, there is an isomorphism (J : ~ ~ ij such that lal = l(Jal as required.

In order to prove that K = ~ or = C we show that each ~ E K satisfies a quadratic equation over ~. For this, consider the continuous function f : C ~ ~ defined by

fez) = I ~2 - (z + z)~ + zzl.


Note here that z + z, zz E ~ 5;; K. Since lim fez) = 00, fez) has a z---+oo

minimum m. The set

s = {z Eel fez) = m}

is therefore nonempty, bounded, and closed, and there is a Zo E S such that Izol :::: Izl for all z E S. It suffices to show that m = 0, because then one has the equation ~2 - (zo + zo)~ + zozo = O.

Assume m > O. Consider the real polynomial

g(x) = x 2 - (zo + zo)x + zozo + s,

where 0 < s < m, with the roots Zl, Zl E Co We have ZlZl = zozo + s, hence Iz1l > Izol and thus

f(zl) > m.

For fixed n EN, consider on the other hand the real polynomial

2n 2n G(x) = [g(x) - s r - (_s)n = IT (x - ai) = IT (x - iii)

i=l i=l

with roots aI, ... , a2n E Co It follows that G(Zl) = 0; say, zl = a1. We may substitute ~ E K into the polynomial

and get

2n G(x)2 = IT{x2 - (ai +iii)X +aiiii)

i=l

2n I G(~)12 = IT f(ai) :::: f(al)m 2n- 1.

i=l

From this and the inequality

I G(~)I :5 1~2 - (zo + zo)~ + zozoln + I - sin = f(zo)n + sn = mn + sn,

it follows that f(a1)m 2n- l :5 (mn + sn)2 and hence

f~l) :5 (1 + (:rr. For n --+ 00 we have f(a1) :5 m, which contradicts the inequality f(al) > m proved before. 0

In view of OSTROWSKI'S theorem, we will henceforth restrict attention to the case of nonarchimedean valuations. In this case it is usually expedient -both with regard to the substance and to practical technique - to work with the exponential valuations v rather than the multiplicative valuations. So let v


be an exponential valuation of the field K. It is canonically continued to an exponential valuation v of the completion K by setting .

v (a) = lim v(an), n-+oo

~

where a = lim an E K, an E K. Observe here that the sequence v(an) n-+oo

has to become stationary (provided a i= 0) because, for n ~ no, one has v(a - an) > v(a), so that it follows from the remark on p. 119

v(an) = Han - a + a) = min{ v(an - a), v(a)} = v(a).

Therefore it follows that

v(K*) = v(K*),

and if v is discrete and normalized, then so is the extension v. In the nonarchimedean case, for a sequence {an}nEN to be a Cauchy sequence, it suffices that an+l - an be a nUllsequence. In fact, v(an - am) ~ minm::;i<n{v(a.i+1 - ai)}. By the same token an infinite series 2:=:oav

converges in K if and only if the sequence of its terms av is a nUllsequence. The following proposition is proved exactly as its analogue, proposition (2.4), in the special case (Q, v p) .

(4.3) Proposition. If 0 s;; K, resp. -0 s;; K, is the valuation ring of v, resp. of v, and p, resp. p, is the maximal ideal, then one has

and, if v is discrete, one has furthermore

Generalizing the p-adic expansion to the case of an arbitrary discrete valuation v of the field K, we have the

(4.4) Proposition. Let R S; 0 be a system of representatives for K = Oil such that 0 E R, and let Jr E 0 be a prime element. Then every x i= 0 in K admits a unique representation as a convergent series

x = Jrm(ao + alJr + a2Jr2 + ... )

where aj E R, ao i= 0, mE Z.


Proof: Let x = nn1 u with U E a*. Since alp ~ alp, the class u mod p has a unique representative ao E R, ao i= 0. We thus have u = ao + n bI ,

for some bl E a. Assume now that ao, ... , an-I E R have been found, satisfying

for some bn E a, and that the aj are uniquely determined by this equation. Then the representative an E R of bn mod na E alp ~ alp is also uniquely determined by u and we have bn = an + n bn+ I, for some bn+ I Ea.

Hence

u = ao + aln + ... + an_Inn- 1 + annn + nn+lbn+l.

In this way we find an infinite series L~=o aJ)n J) which is uniquely determined by u. It converges to u because the remainder term nn+ I bn+ I tends to zero. 0

In the case of the field of rational numbers Q and the p-adic valuation vp

with its completion Qp' the numbers 0,1, ... , p - 1 form a system of representatives R for the residue class field 7/.,1 p7/., of the valuation, and we get back the representation of p -adic numbers which has already been discussed in § 2:

x = pn1(ao + alP + a2p2 + ... ), where ° :::: aj < p and mE7/.,.

In the case of the rational function field k(t) and the valuation vp attached to a prime ideal p = (t - a) of k[t] (see §2), we may take as a system of representatives R the field of coefficients k itself. The completion then turns out to be the field of formal power series k((x)), x = t - a, consisting of all formal Laurent series

f(t) = (t - a)n1(ao + al (t - a) + a2(t - a)2 + ... ), with aj E k and mE7/.,. The motivating analogy of the beginning of this chapter, between power series and p-adic numbers, thus appears as two special instances of the same concrete mathematical situation.

In § 1 we identified the ring 7/.,p of p-adic integers as being the projective limit ¥!!! 7/., I pn 7/.,. We obtain a similar result in the general setting of

n valuation theory. To explain this, let K be complete with respect to a discrete valuation. Let a be the valuation ring with the maximal ideal p. We then have for every n ::::: 1 the canonical homomorphisms

a ~ a/pn


and

This gives us a homomorphism

into the projective limit

00

~ o/pn = {(Xn) E IT O/pn I An(Xn+l) = Xn} . n n=l

Considering the rings o/pn as topological rings, for the discrete topology, gives us the product topology on IT~=1 o/pn, and the projective limit ~ o/pn becomes a topological ring in a canonical way, being a closed

n subset of the product (see chap. IV, § 2).

(4.5) Proposition. The canonical mapping

0----+ lim o/pn +-

n

is an isomorphism and a homeomorphism. The same is true for the mapping

0* ----+ ~ 0*/ U(n) .

n

Proof: The map is injective since its kernel is n~1 pn = (0). To prove surjectivity, let p = 7rO and let R ~ 0, R 30, be a system of representatives of o/p. We saw in the proof of (4.4) (and in fact already in (1.2» that the elements a mod pn E o/pn can be given uniquely in the form

a == ao + a17r + ... + an_17rn-1 mod pn,

where ai E R. Each element S E ~ o/pn is therefore given by a sequence of sums n

n-l I 2 Sn = ao + al7r + ... + an-l7r , n = , , ... ,

with fixed coefficients ai E R, and it is thus the image of the element x = lim Sn = L:oav7r v EO.

n---+oo

The sets Pn = ITv>n o/pv form a basis of neighbourhoods of the zero

element of IT:1 o/pv. Under the bijection

o ----+ ~ o/pv v


the basis of neighbourhoods pn of zero in 0 is mapped onto the basis of neighbourhoods Pn n ¥!!! o/pv of zero in ¥!!! o/pv. Thus the bijection is

v v a homeomorphism. It induces an isomorphism and homeomorphism on the group of units

One of our chief concerns will be to study the finite extensions L I K of a complete valued field K. This means that we have to turn to the question of factoring algebraic equations

f(x) = anxn + an_lxn- 1 + ... + ao = 0

over complete valued fields. For this, Hensel's seminal "lemma" is of fundamental importance. Let K again be a field which is complete with respect to a nonarchimedean valuation I I. Let 0 be the corresponding valuation ring with maximal ideal p and residue class field K = o/p. We call a polynomial f(x) = ao + alx + ... + anxn E o[x] primitive if f(x) ¢. 0 mod p, i.e., if

If I = maxI laol, ... , lanl} = 1.

(4.6) Hensel's Lemma. If a primitive polynomial f(x) E o[x] admits modulo p a factorization

f(x) == g(x)h (x) mod p

into relatively prime polynomials g, h E K [x], then f (x) admits a factorization

f(x) = g(x)h(x)

into polynomials g, h E o[x] such that deg(g) = deg(g) and

g(x) == g(x) mod p and hex) == hex) mod p.

Proof: Let d = deg(f) , m = deg(g), hence d - m ::: deg(h). Let go, ho E o[x] be polynomials such that go == g mod p, ho == h mod p and deg(go) = m, deg(ho) .:::: d - m. Since (g, h) = 1, there exist polynomials a(x), b(x) E o[x] satisfying ago + bho == 1 mod p. Among the coefficients of the two polynomials f - goho and ago + bho - 1 E p[x] we pick one with minimum value and call it 7i .


Let us look for the polynomials g and h in the following form:

g = go + Pl:rr + P2:rr 2 + ... , h = ho + ql:rr + q2:rr 2 + ... ,

where Pi, qi E o[x] are polynomials of degree < m, resp. ::: d - m. We then determine successively the polynomials

gn-l = go + Pl:rr + ... + Pn_l:rrn- 1,

hn- 1 = ho + ql:rr + ... + qn_l:rr n- 1 ,

in such a way that one has

I == gn-lhn-I mod :rrn.

Passing to the limit as n --+ 00, we will finally obtain the identity I = gh. For n = 1 the congruence is satisfied in view of our choice of :rr. Let us assume that it is already established for some n ::: 1. Then, in view of the relation

gn = gn-I + Pn:rrn , hn = hn- 1 + qn:rrn,

the condition on gn, hn reduces to

1- gn-lhn-l == (gn-lqn + hn_1Pn):rrn mod :rrn+1.

Dividing by :rrn, this means

gn-lqn + hn-1Pn == goqn + hOPn == In mod:rr,

where In = :rr-n(f - gn-lhn-l) E o[x]. Since goa + hob == 1 mod :rr, one has

goaln + hobln == In mod :rr .

At this point we would like to put qn = aln and Pn = bin, but the degrees might be too big. For this reason, we write

b(x)ln(x) = q(x)go(x) + Pn(x),

where deg(Pn) < deg(go) = m. Since go == g mod p and deg(go) = deg(g), the highest coefficient of go is a unit; hence q(x) E o[x] and we obtain the congruence

go(aln + hoq) + hoPn == In mod:rr .

Omitting now from the polynomial aln + hoq all coefficients divisible by :rr, we get a polynomial qn such that goqn + hoPn == In mod:rr and which, in view of deg(fn) ::: d, deg(go) = m and deg(hoPn) < (d - m) + m = d, has degree ::: d - m as required. 0


Example: The polynomial x p-l -1 E Z p [x] splits over the residue class field Zp/ pZp = IF p into distinct linear factors. Applying (repeatedly) Hensel's lemma, we see that it also splits into linear factors over Z p. We thus obtain the astonishing result that the field Q2p of p-adic numbers contains the (p -l)-th roots of unity. These, together with 0, even form a system of representatives for the residue class field, which is closed under multiplication.

(4.7) Corollary. Let the field K be complete with respect to the nonarchimedean valuation I I. Then, for every irreducible polynomial I (x) ao + alx + ... + anxn E K[x] such that aoan #- 0, one has

III = max{ laol, Ian I}.

In particular, an = 1 and ao E 0 imply that I E o[x].

Proof: After multiplying by a suitable element of K we may assume that I E o[x] and III = 1. Let ar be the first one among the coefficients ao, ... , an such that lar I = 1. In other words, we have

I(x) == xr (ar + ar+lX + ... + anxn- r ) mod p.

If one had max { I ao I, I an I} < 1, then ° < r < n and the congruence would contradict Hensel's lemma. 0

From this corollary we can now deduce the following theorem on extensions of valuations.

(4.8) Theorem. Let K be complete with respect to the valuation I I. Then I I may be extended in a unique way to a valuation of any given algebraic extension L I K. This extension is given by the formula

when L IK has finite degree n. In this case L is again complete.

Proof: If the valuation I I is archimedean, then by Ostrowski's theorem, K = lR or Co We have NqlR(z) = ZZ = IzI2 and the theorem is part of classical analysis. So let I I be nonarchimedean. Since every algebraic extension L I K is the union of its finite subextensions, we may assume that the degree n = [L : K] is finite.


Existence of the extended valuation: let 0 be the valuation ring of K and (J

its integral closure in L. Then one has

The implication a E (J ~ NLIK(a) E 0 is evident (see chap. I, § 2, p. 12). Conversely, let a E L* and NLIK(a) E o. Let

f(x) = xd + ad_txd-t + ... + ao E K[x]

be the minimal polynomial of a over K. Then N L IK (a) = ±ag' EO, so that laol ::: 1, i.e., ao E o. By (4.7) this gives f(x) E o[x], i.e., a E O.

For the function lal = -V'INLIK(a)l, the conditions lal = 0 {:=} a = 0 and la.B1 = lall.B1 are obvious. The strong triangle inequality

la +.BI ::: max{ lal, I.BI} reduces, after dividing by a or .B, to the implication

lal ::: 1 ===} la + 11::: 1,

and then, by (*), to a E (J ~ a + 1 E (J, which is trivially true. Thus the formula lal = -V'INLIK (a) I does define a valuation of L and, restricted to K, it clearly gives back the given valuation. Equally obviously it has (J

as its valuation ring.

Uniqueness of the extended valuation: let I I' be another extension with valuation ring (J'. Let~, resp. ~', be the maximal ideal of (J, resp. (Jf. We show that (J £; (J'. Let a E (J " (J' and let

f(x) = xd + atxd- t + ... + ad

be the minimal polynomial of a over K. Then one has at, ... , ad E 0 and a-t E ~', hence 1 = -ala- l - ... - ad(a-t)d E \p', a contradiction. This shows the inclusion (J £; (J'. In other words, we have that la I ::: 1 ~ la I' ::: 1 and this implies that the valuations I I and I I' are equivalent. For if they were not, then the approximation theorem (3.4) would allow us to find an a E L such that I a I ::: 1 ~ la I' > 1. Thus I I and I I' are equal because they agree on K.

The fact that L is again complete with respect to the extended valuation is deduced from the following general result. D

(4.9) Proposition. Let K be complete with respect to the valuation I I and let V be an n-dimensional nonned vector space over K. Then, for any basis Vt, •.. , Vn of V the maximum nonn

IIXtVl + ... +xnvnll = max{ lXII, ••. , IXnl}


is equivalent to the given nonn on V. In particular, V is complete and the isomorphism

is a homeomorphism.

Proof: Let VI, ... , Vn be a basis and II II be the corresponding maximum nonn on V. It suffices to show that, for every norm I I on V, there exist constants p, pi > 0 such that

pllxlI.:::: Ixl.:::: P'llxll forall x E V.

Then the norm I I defines the same topology on V as the norm II II, and we obtain the topological isomorphism K n -+ V, (XI, ... , xn) f-+

XI VI + ... +xnvn . In fact, II II is transformed into the maximum norm on Kn.

For pi we may obviously take I VI I + ... + I Vn I. The existence of p is proved by induction on n. For n = 1 we may take p = I vII. Suppose that everything is proved for (n - I)-dimensional vector spaces. Let

Vi = K VI + ... + K Vi-I + K Vi+! + ... + K Vn ,

so that V = Vi + K Vi. Then Vi is complete with respect to the restriction of I I by induction, hence it is closed in V. Thus Vi + Vi is also closed. Since 0 1. U7=1 (Vi + Vi), there exists a neighbourhood of 0 which is disjoint from U7=1 (Vi + vd, i.e., there exists p > 0 such that

I Wi + Vi I :::: p for all Wi E Vi and all i = 1, ... , n.

For X = XI VI + ... + XnVn i 0 and IXr I = max{lxi I}, one finds

-I I XI Xn I IXr X I = - VI + ... + Vr + ... + - Vn :::: p, Xr Xr

so that one has Ixi :::: plxrl = plixli. o

The fact that an exponential valuation V on K associated with I I extends uniquely to L is a trivial consequence of theorem (4.8). The extension W is given by the formula

if n = [L : K] < 00.


Exercise 1. An infinite algebraic extension of a complete field K is never complete.

Exercise 2. Let Xo, X I, ... be an infinite sequence of unknowns, p a fixed prime n 11-1

number and Wn = xt + pXf + ... + pnxn, n :::: O. Show that there exist polynomials So, SI, ... ; Po, PI, ... E Z[Xo, X I, ... ; Yo, YI , ... ] such that

Wn(SO,SI, ... ) = Wn(XO,X I , ... ) + Wn(YO,YI , ... ),

Wn(PO,PI , ... ) = Wn(XO,X I , ... ). Wn(YO,YI , ... ).

Exercise 3. Let A be a commutative ring. For a = (aO,al, ... ), b = (bo,b l , ... ), ai, bi E A, put

a + b = (So(a, b), SI (a, b), ... ), a· b = (Po(a, b), PI (a, b), ... ).

Show that with these operations the vectors a = (ao, ai, ... ) form a commutative ring W (A) with 1. It is called the ring of Witt vectors over A.

Exercise 4. Assume pA = O. For every Witt vector a = (aO,al, ... ) E W(A) consider the "ghost components"

n n-l a(n) = Wn(a) = at + paf + ... + pnan

as well as the mappings V, F : W (A) -+ W (A) defined by

Va = (O,aO,al, ... ) and Fa = (at,af, ... ),

called respectively "transfer" ("Verschiebung" in German) and "Frobenius". Show that

(Va)(n) = pa(n-l) and a(n) = (Fa)(n) + pnan .

Exercise 5. Let k be a field of characteristic p. Then V is a homomorphism of the additive group of W(k) and F is a ring homomorphism, and one has

VFa = FVa = pa.

Exercise 6. If k is a perfect field of characteristic p, then W(k) is a complete discrete valuation ring with residue class field k.

§ 5. Local Fields

Among all complete (nonarchimedean) valued fields, those ansmg as completions of a global field, i.e., of a finite extension of either <Q or IF pet), have the most eminent relevance for number theory. The valuation on such a completion is discrete and has a finite residue class field, as we shall see shortly. In contrast to the global fields, all fields which are complete with respect to a discrete valuation and have a finite residue class field are called local fields. For such a local field, the normalized exponential valuation is denoted by vp, and I I p denotes the absolute value normalized by

Ix Ip = q-vp(x),

where q is the cardinality of the residue class field.

§ 5. Local Fields 135

(5.1) Proposition. A local field K is locally compact. Its valuation ring a is compact.

Proof: By (4.5) we have a ~ ~ a/pn, both algebraically and topologically. Since pV /pv+I ~ a/p (see (3.9», the rings a/pn are finite, hence compact. Being a closed subset of the compact product n~=1 a /pn , it follows that the projective limit ~ a/pn, and thus a, is also compact. For every a E K, the set a + a is an open, and at the same time compact neighbourhood, so that K is locally compact. 0

In happy concord with the definition of global fields as the finite extensions of Q and IF p (t), we now obtain the following characterization of local fields.

(5.2) Proposition. The local fields are precisely the finite extensions of the fields Qp and IF p«t».

Proof: A finite extension K of k = Qp or k = IF p«t» is again complete, by (4.8), with respect to the extended valuation lal = ytINKlk(a) I, which itself is obviously again discrete. Since K Ik is of finite degree, so is the residue class field extension K IlF p, for if Xl, ... , X n E K

are linearly independent, then any choice of preimages xl, ... , Xn E K is linearly independent over k. Indeed, dividing any nontrivial k-linear relation AIXI + ... + AnXn = 0, Ai E k, by the coefficient Ai with biggest absolute value, yields a linear combination with coefficients in the valuation ring of k with 1 as i -th coefficient, from which we obtain a nontrivial relation x: I x I + ... + x: nX n = 0 by reducing to K. Therefore K is a local field.

Conversely, let K be a local field, v its discrete exponential valuation, and p the characteristic of its residue class field K. If K has characteristic 0, then the restriction of v to Q is equivalent to the p-adic valuation vp of Q because v(p) > o. In view of the completeness of K, the closure of Q in K is the completion of Q with respect to v p' in other words Q p £;; K. The fact that K IQp is of finite degree results from the local compactness of the vector space K, by a general theorem of topological linear algebra (see [18], chap. I, § 2, nO 4, tho 3), but it also follows from (6.8) below. If on the other hand the characteristic of K is not equal to zero, then it has to equal p. In this case we find K = K«t», for a prime element t of K (see p.127), hence lFp«t» £;; K. In fact, if K = lFp(a) and p(X) E lFp[X] £;; K[X] is the minimal polynomial of a over IF p' then, by Hensel's lemma, p(X) splits over K into linear factors. We may therefore view K as a subfield of K, and then the elements of K tum out to be, by (4.4), the Laurent series in t with coefficients in K. 0


Remark: One can show that a field K which is locally compact with respect to a nondiscrete topology is isomorphic either to ~ or C, or to a finite extension of Qp or IF p«(t», i.e., to a local field (see J137], chap. I, § 3).

We have just seen that the local fields of characteristic p are the power series fields lFq«(t», with q = pt. The local fields of characteristic 0, i.e., the finite extensions K IQ p of the fields of p-adic numbers Qp' are called p-adic number fields. For them one has an exponential function and a logarithm function. In contrast to the real and complex case, however, the former is not defined on all of K, whereas the latter is given on the whole multiplicative group K*. For the definition of the logarithm we make use of the following fact.

(5.3) Proposition. The multiplicative group of a local field K admits the decomposition

K* = (rr) x /1-q-l x U(I).

Here rr is a prime element, (rr) = {rrk IkE Z}, q = #K is the number of elements in the residue class field K = o/p, and U(I) = 1 + p is the group of principal units.

Proof: For every a E K*, one has a unique representation a = rrnu with nEZ, u E 0* so that K* = (rr) x 0*. Since the polynomial X q - 1 - 1 splits into linear factors over K by Hensel's lemma, 0* contains the group /1-q-l of (q - l)-th roots of unity. The homomorphism 0* -+ K*, U f--+ U mod p, has kernel U(I) and maps /1-q-1 bijectively onto K*. Hence 0* = /1-q-l x U(I). 0

(5.4) Proposition. For a p-adic number field K there is a uniquely determined continuous homomorphism

log: K* -+ K

such that log p = ° which on principal units (1 + x) E U (I) is given by the series

x 2 x 3 log(1 + x) = x - - + - - ....

2 3

Proof: By §4, we can think of the p-adic valuation vp of Qp as extended to K. Observing that vp(x) > 0, so that c = pVp(x) > 1, and pVp(v) :::: v,


giving vp(v) ~ lIn v (with the usual logarithm), we compute the valuation of np

the terms x v I v of the series,

(Xv) Inc lnv In(cVlv) v - = vv (x) - v (v) > v - - - = . p v p p - lnp lnp lnp

This shows that XV Iv is a nullsequence, i.e., the logarithm series converges. It defines a homomorphism because

log( (1 + x)(1 + y») = 10g(1 + x) + 10g(1 + y)

is an identity of formal power series and all series in it converge provided l+x,l+YEU(1).

For every a E K*, choosing a prime element rr, we have a unique representation

a = rrvp(a)w(a)(a) ,

where vp = evp is the normalized valuation of K, w(a) E /lq-l, (a) E U(1).

As suggested by the equation p = rrew(p)(p), we define logrr = -~ log (p) and thus obtain the homomorphism log: K* --+ K by

loga = vp(a)logrr + log (a).

It is obviously continuous and has the property that log p = O. If J... : K* --+ K is any continuation of log : U(1) --+ K such that J...(p) = 0, then we

find that J...(~) = _1-1J...(~q-l) = 0 for each ~ E /lq-l. It follows that q-

o = eJ...(rr) + J...«(p) = eJ...(rr) + log (p), so that J...(rr) = logrr, and thus A(a) = vp(a)A(rr) +A«(a) = vp(a) logrr + log (a) = loga, for all a E K*. log is therefore uniquely determined and independent of the choice of rr.

o

(5.5) Proposition. Let KIQ p be a p-adic number field with valuation ring o and maximal ideal p, and let po = pe. Then the power series

x 2 x 3 exp(x) = 1 + x + - + - + ...

2! 3! and

Z2 z3

log(l + z) = z - "2 + "3 - ... ,

yield, for n > ~l' two mutually inverse isomorphisms (and homeomorp-

phisms) exp

pn ~ U(n).

log

We prepare the proof by the following elementary lemma.

l38 Chapter II. The Theory of Valuations

(5.6) Lemma. Let v = L~ =0 ai pi , ° S ai < p, be the p -adic expansion of the natural number v EN. Then

Proof: Let [c] signify the biggest integer S c. Then we have

[vip] = al + a2P + ... + arpr-l,

[vlp2] = a2 + ... + arpr-2,

ar'

Now we count how many numbers 1,2, ... , v are divisible by p, and then by p2, etc. We find

vp(v!) = [vip] + ... + [vlpr] = al + (p + l)a2 + ... + (pr-l + ... + l)ar

and hence r

(p -1)vp(v!) = (p -l)al + (p2 -1)a2 + ... + (pr -1)ar = L ai(pi -1). i=O

o

Proof of (5.5): We again think of the p-adic valuation vp of Qp as being extended to K. Then vp = evp is the nonnalized valuation of K. For every natural number v > 1, one has the estimate

vp(v) 1 --<--, v-l-p-l

for if v = pa vo , with (vo, p) = 1 and a > 0, then

vp(v) a a 1 a 1 = < < --.

v-I pa vo - 1 - pa - 1 p - 1 pa-l + ... + p + 1 - p - 1

For vp(z) > _1-1' z i= 0, i.e., vp(z) > ~1' this yields p- p-

vp(Zv) _ vp(z) = (v - l)vp(z) - vp(v) > (v _1)(_1 __ Vp(V») ::: 0, v p-l v-I

and thus vp(log(l + z» = vp(z). For n > ~1' log therefore maps u(n) p-

into pn. For the exponential series L~=o x v I v !, we compute the valuations

vp(XV Iv!) as follows. Writing, for v > 0,

v = ao + al P + ... + ar pr ,Os ai < p,


we get from (5.6) that 1 r. 1

vp (v!) = -- L ai (pi - 1) = -- ( v - (ao + al + ... + ar ) ) . p - 1 i=O P - 1

Putting s v = ao + ... + ar this becomes

vp(xv) = vVp(x) _ v - Sv = v (Vp(x) __ 1_) + ~. v! p-l p-l p-l

For vp(x) > ~1' i.e., vp(x) > ~1' this implies the convergence of the p- p-

exponential series. If furthermore x i= 0 and v > 1, then one has

( xv) v-I Sv - 1 Sv - 1 v - - v (x) = (v - l)v (x) - -- + -- > -- ~ O. p v! p p p-l p-l p-l

Therefore vp(exp(x) -1) = vp(x), i.e., for n > ~1' exp maps the group pn p-

into u(n). Furthermore, one has for vp(x), vp(z) > ~1 that p-

explog(1 + z) = 1 + z and logexp x = x,

for these are identities of formal power series and all of the series converge. This proves the proposition. 0

For an arbitrary local field K, the group of principal units U(l) is a Z -module (where p = char(K» in a canonical way, i.e., for every 1 +x E U(l) and every Z E ZP' one has the power (1 + x)Z E U(l). This is a consequence of the fact that U(l) ju(n+l) has order qn for all n (where q = #ojp - the reason for this is that U(i) jU(i+l) ~ ojp, by (3.10), so that U(1) ju(n+l) is a ZjqnZ-module) and of the formulas

U(l)= ~ U(l)jU(n+l) and Zp= ~ ZjqnZ. n n

This obviously extends the Z-module structure of U(l). The function

fez) = (1 + x)Z

is continuous because the congruence z == z' mod qn Z p implies (1 + x)Z == (1 + x/ mod u(n+l), so that the neighbourhood z + qnzp of z is mapped to the neighbourhood (1 +x)zu(n+l) of fez). In particular, (1 +x)Z may be expressed as the limit

(1 + x)Z = lim (1 + x)Z; j-+oo

of ordinary powers (1 + x)Z;, Zj E Z, if z = .lim Zj. 1-+00

After this discussion we can now determine explicitly the structure of the locally compact multiplicative group K* of a local field K.


(5.7) Proposition. Let K be a local field and q = pi the number of elements in the residue class field. Then the following hold.

(i) If K has characteristic 0, then one has (both algebraically and topologically)

K* ~ 71.. E9 Z/(q - 1)71.. E9 Z/paZ E9 Z~,

where a ::: 0 and d = [K : Qp].

(ii) If K has characteristic p, then one has (both algebraically and topologically)

Proof: By (5.3) we have (both algebraically and topologically)

K* = (Jl') X JLq-I x U(I) ~ 71.. E9 Z/(q - 1)71.. E9 U(I) .

This reduces us to the computation of the Zp-module U(I).

(i) Assume char(K) = O. For n sufficiently big, (5.5) gives us the isomorphism

log: U(n) ---+ pn = Jl'nO ~ o.

Since log, exp, and f(z) = (1 + x)Z are continuous, this is a topological isomorphism of Zp-modules. By chap.I, (2.9), 0 admits an integral basis aI, ... , ad over 7l..p , i.e., 0 = 7l..pal E9 ... E9 7l..pad ~ z~. Therefore

u(n) ~ 71..%. Since the index (U(I) : u(n» is finite and u(n) is a finitely

generated Zp-module of rank d, so is U(I). The torsion subgroup of U(I) is the group JLpa of roots of unity in K of p-power order. By the main theorem on modules over principal ideal domains, there exists in U(I) a free, finitely generated, and therefore closed, 71.. p -submodule V of rank d such that

U(I) = JLpa x V ~ 71../ paZ E9 Z~,

both algebraically and topologically.

(ii) If char(K) = p, we have K ~ IF q«t» (see p. 127) and

U(I) = 1 +p = 1 +tIFq[[t]].

The following argument is taken from the book [79] of K. [WASAWA.

Let WI, ..• , WI be a basis of IF q IIF p. For every natural number n relatively prime to p we consider the continuous homomorphism

g . ...,,1 --+ U(n) n· ~p ,

I gn(aI, ... , al) = n (1 + witn)ai •

i=I

§ 5. Local Fields

This function has the following properties. If m = nps, s :::: 0, then

(1) U(m) = gn(pSZ£)U(m+l)

and, for a = (aI, ... , af) E Z£,

(2) a 1. pZ£ {=::} gn(pSa) 1. U(m+I).

Indeed, for w = "LLI biWi E lFq, bi E Z, bi == ai mod p, we have

f gn(a) == TI (1 + witn)hi == 1 + wtn mod pn+1

i=1

and hence, since we are in characteristic p,

gn(pSa) = gn(a)Ps == 1 + w Ps tm mod pm+l.

141

As a varies over the elements of Z £ ' w, and thus also wPs , varies over the elements of IF q , and we get (l). Furthermore one has gn (pS a) == 1 mod pm+ I {=::}

W = 0 {=::} bi == 0 mod p, for i = 1, ... , f {=::} ai == 0 mod p, for i = 1, ... , f {=::} a E pZ£, and this amounts to (2).

We now consider the continuous homomorphism of Z P -modules

g = TI gn: A = TI Z£ --+ U(l) , (n, p)=1 (n, p)=1

where the product TI(n,p)=1 Z£ is taken over all n :::: I such that (n, p) = 1,

each factor being a copy of Z{ Observe that the product g(~) = TI gn(an)

converges because gn(an) E u(n). Let m = nps, with (n, p) = 1, be any

natural number. As gn(Z£) S; g(A), it follows from (1) that each coset of u(m) / u(m+l) is represented by an element of g(A). This means that g(A)

is dense in U(l). Since A is compact and g is continuous, g is actually surjective.

On the other hand, let ~ = ( ... , an, ... ) E A, ~ =j:. 0, i.e., an =j:. 0 for some n. Such an an is of the form an = pS f3n with s = sean) :::: 0, and

f3n E z£ " pz£. It now follows from (2) that

gn (an) E U(m), gn (an) 1. U(m+l) for m = mean) = nps .

Since the n are prime to p, all the mean) have to be distinct, for all an =j:. O. Let n be the natural number, prime to p and such that an =j:. 0, which satisfies mean) < mean')' for all n' =j:. n such that an' =j:. O. Then one has, for all n' =j:. n, that

gn,(an,) E U(m+l) where m = mean) < mean')'

Consequently g(~) == gn(an) ¢. 1 mod U(m+I) ,

and so g(~) =j:. 1. This shows the injectivity of g. Since A = Z;, this proves the claim (ii). 0


(5.8) Corollary. If the natural number n is not divisible by the characteristic of K, then one finds the following indices for the subgroups of n-th powers KM and un in the multiplicative group K* and in the unit group U:

(K* : K*n) = n(U : Un) = ~#J.Ln(K). Inlp

Proof: The first equality is a consequence of K* = (n) x U. By (5.7), we have

U ~ J.L(K) x Z~, resp. U ~ J.L(K) x Z~ , when char (K) = 0, resp. p > 0. From the exact sequence

1 ---+ J.Ln(K) ---+ J.L(K) ~ J.L(K) ---+ J.L(K)/J.L(K)n ---+ 1,

one has #J.Ln(K) = #J.L(K)/J.L(K)n. When char(K) = 0, this gives:

(U : Un) = #J.Ln(K)#(Zp/nZp)d = #J.Ln(K)pdvp(n) = #J.Ln(K)/lnlp,

and when char(K) = p one gets simply (U : un) = #J.Ln(K) = #J.Ln(K)/lnlp because (n, p) = 1, i.e., nZp = Zp. 0

Exercise 1. The logarithm function can be continued to a continuous homomorphism log : Q; ~ Q, and the exponential function to a continuous homomorphism

exp : p I~P ~ «J;" where p I~P = {x E Qp I vp(x) > I~P} and vp is the unique extension of the nonnalized valuation on Qp.

Exercise 2. Let K IQp be a p-adic number field. For 1 + x E U(l) and Z E Zp one has

00

(l + x)Z = L e)x v •

v=O

The series converges even for x E K such that vp(x) > ~1' p-

Exercise 3. Under the above hypotheses one has

(l + x)Z = exp(z log(l + x» and log(l + x)Z = z log(l + x).

Exercise 4. For a p-adic number field K, every subgroup of finite index in K* is both open and closed.

Exercise 5. If K is a p-adic number field, then the groups K*n, for n EN, fonn a basis of neighbourhoods of 1 in K*.

Exercise 6. Let K be a p-adic number field, vI' the nonnalized exponential valuation of K, and dx the Haar measure on the locally compact additive group K, scaled so that fo dx = 1. Then one has vp(a) = fao dx. Furthennore,

JU) = ( I(x) IdXI lK"-.{o) x I'

is a Haar measure on the locally compact group K*.

§ 6. Henselian Fields 143

§ 6. Henselian Fields

Most results on complete valued fields can be derived from Hensel's lemma alone, without the full strength of completeness. This lemma is valid in a much bigger class of nonarchimedean valued fields than the comJ?lete ones. For example, let (K, v) be a nonarchimedean valued field and (K, fj) its completion. Let 0, resp. 0, be the valuation rings of K, resp. K. We then consider the separable closure K v of K in K, and the valuation ring Ov ~ K v with maximal ideal Pv, which is associated to the restriction of fj to K v,

K ~ Kv ~ K, 0 ~ Ov ~ o. Then Hensel's lemma holds in the ring Ov as well as in the ring 0 even though Kv will not, as a rule, be complete. When Kv is algebraically closed in K - hence in particular char(K) = 0 - this is immediately obvious (otherwise it follows from (6.6) and §6, exercise 3 below). Indeed, by (4.3) we have

O/p = ov/Pv = o/p, and if a prumtlve polynomial I(x) E ov[x] splits over ov/Pv into relatively prime factors g(x),h(x), then we have by Hensel's lemma (4.6) a factorization in 0

I(x) = g(x)h(x)

such that g == g modp, h == h modp, deg(g) = deg(g). But this factorization already takes place over Ov once the highest coefficient of g is chosen to be in o~, because the coefficients of I, and therefore also those of g and h are algebraic over K.

The valued field K v is called the henselization of the field K with respect to v. It enjoys all the relevant algebraic properties of the completion K, but offers the advantage of being itself an algebraic extension of K which can also be obtained in a purely algebraic manner, without the analytic recourse to the completion (see §9, exercise 4). The consequence is that taking the henselization of an infinite algebraic extension L I K is possible within the category of algebraic extensions. Let us define in general:

(6.1) Definition. A henselian field is a field with a nonarchimedean valuation v whose valuation ring 0 satisfies Hensel's lemma in the sense of (4.6). One also calls the valuation v or the valuation ring 0 henselian.


(6.2) Theorem. Let K be a henselian field with respect to the valuation 1 I. Then 1 1 admits one and only one extension to any given algebraic extension L 1 K. It is given by

if L IK has finite degree n. In any case, the valuation ring of the extended valuation is the integral closure of the valuation ring of K in L.

The proof of this theorem is verbatim the same as in the case of a complete field (see (4.8». What is remarkable about our current setting is that, conversely, the unique extendability also characterizes henselian fields. In order to prove this, we appeal to a method which allows us to express the valuations of the roots of a polynomial in terms of the valuations of the coefficients. It relies on the notion of Newton polygon, which arises as follows.

Let v be an arbitrary exponential valuation of the field K and let

I(x) = ao + alX + ... + anxn E K[x]

be a polynomial satisfying aoan =I O. To each term aixi we associate a point (i, v(aj) E ]R2, ignoring however the point (i, (0) if ai = o. We now take the lower convex envelope of the set of points

{ (0, v(ao», (1, v(al», ... , (n, v(an»} .

This produces a polygonal chain which is called the Newton polygon of I(x).

(0, v (ao))

The polygon consists of a sequence of line segments Sl, S2, ... whose slopes are strictly increasing, and which are subject to the following


(6.3) Proposition. Let I(x) = ao + aIX + ... + anxn, aoan =j:. 0, be a polynomial over the field K, v an exponential valuation of K, and w an extension to the splitting field L of I.

If (r, v (ar )) *+ (s, v(as)) is a line segment of slope -m occurring in the Newton polygon of I, then I(x) has precisely S - r roots aI, ... , as- r of value

w(aI) = ... = W(as- r ) = m.

Proof: Dividing by an only shifts the polygon up or down. Thus we may assume that an = 1. We number the roots aI, ... , an E L of I in such a way that

w(aI) = .. .

w(as,+I) = .. .

w(as,) = mI,

w(aS2 ) = m2,

where mi < m2 < ... < mt+I. Viewing the coefficients ai as elementary symmetric functions of the roots a j, we immediately find

v(an) = v(1) = 0,

v(an-I) ~ mint w(ad} = mI, I

v(an-2) > mint w(aiaj)} = 2mI, I, ]

v(an- s,) = . min {w(ai, ... ais ,)} = SImI, 11, ... , lSI

the latter because the value of the term al ... as, is smaller than that of all the others,

v(an-s,-I) > . mi!1 {w(ai, ... ais,+')} = sImI + m2, q, ... , I s,+'

v(an- s,-2) > . min {w(ai, .. . ais , +2)} = sImI + 2m2, I" ... , I s,+2

v(an- s2 ) = . m~ {w(ai, ... aiS2 )} II, ... , IS2

and so on. From this result one concludes that the vertices of the Newton polygon, from right to left, are given by

(n,O), (n - sI,sImI), (n - S2,sImi + (S2 - sI)m2),


The slope of the extreme right-hand line segment is O-slml

n - (n - sd =-ml,

and, proceeding further to the left,

(simi + ... + (Sj - sj_l)mj) - (simi + ... + (Sj+1 - sj)mj+l)

(n - Sj) - (n - Sj+l)

o

We emphasize that, according to the preceding proposition, the Newton polygon consists of precisely one segment if and only if the roots ai, ... , an of f all have the same value. In general, f (x) factors into a product according to the slopes -mr < ... < -m I,

where

r

f(x) = an TI h(x), j=1

fj(X) = TI (x - ai). w(cxi)=mJ

Here the factor fj corresponds to the (r - j + l)-th segment of the Newton polygon, whose slope equals minus the value of the roots of fj.

(6.4) Proposition. If the valuation v admits a unique extension w to the splitting field L of f, then the factorization

r

f(x) = an TI hex) j=1

is defined already over K, i.e., hex) = TIW(cxi)=m/x - ai) E K[x].

Proof: We may clearly assume that an = 1. The statement is obvious when f(x) is irreducible because then one has ai = O"ial for some O"i E G(LIK), and since, for any extension w of v, w OO"i is another one, the uniqueness implies that w(ai) = W(O"ial) = ml, hence fl(x) = f(x).

The general case follows by induction on n. For n = 1 there is nothing to show. Let p(x) be the minimal polynomial of al and g(x) = f(x)/ p(x) E

K[x]. Since all roots of p(x) have the same value ml, p(x) is a divisor of fl(x). Let gl(x) = /J(x)/p(x). The factorization of g(x) according to the slopes is

r

g(x) = gl(x) TI fj(X). j=2

Since deg(g) < deg(f) , it follows that fj (x) E K [x] for all j = 1, ... , r. 0


If the polynomial f is irreducible, then, by the above factorization result, there is only one slope, i.e., the Newton polygon consists of a single segment. The values of all coefficients lie on or above this line segment and we get the

(6.5) Corollary. Letf(x) =aO+alx+···+anxn E K[x] bean irreducible polynomial with an =1= O. Then, if I I is a nonarchimedean valuation of K with a unique extension to the splitting field, one has

If I = max{ laol, lanl} .

In (4.7) we deduced this result for complete fields from Hensel's lemma and thus obtained the uniqueness of the extended valuation. Here we obtain it, by contrast, as a consequence of the uniqueness of the extended valuation. We now proceed to deduce Hensel's lemma from the unique extendability.

(6.6) Theorem. A nonarchimedean valued field (K, I I) is henselian if and only if the valuation I I can be uniquely extended to any algebraic extension.

Proof: The fact that a henselian valuation I I extends uniquely was dealt with in (6.2). Let us assume conversely that I I admits one and only one extension to any given algebraic extension. We first show:

Let f(x) = ao + alx + ... + anxn E o[x] be a primitive, irreducible polynomial such that aoan =1= 0, and let l(x) = f(x) mod P E K[X]. Then we have deg(J) = 0 or deg(J) = deg(f), and we find

l(x) = a ip(x)m,

for some irreducible polynomial ip(x) E K[X] and a constant a. As f is irreducible, the Newton polygon is a single line segment and thus

If I = max{laol, Ian I}. We may assume that an is a unit, because otherwise the Newton polygon is a segment which does not lie on the x-axis and this means that l(x) = ao.

Let L I K be the splitting field of f (x) over K and 0 the valuation ring of the unique extension I I to L, with maximal ideal Sfl. For an arbitrary K-automorphism (f E G = G(LIK), we have l(fal = lal for all a E L, because I I and the composite I I 0 (f extend the same valuation. This shows that (fO = 0, (fSfl = Sfl. If a is a zero of f(x) and f.-L its multiplicity, then (fa E 0 for all (f E G. Indeed, if a fj. 0, then TIcr l(fal JL = I TIcr (fal JL > 1 would imply that the constant coefficient ao could not belong to o. Thus every (f E G induces a K-automorphism a of 0 jSfl, and the zeroes (fa = aa


of 1 (x) are all conjugate over K. It follows that 1 (x) = acp (x)m, if cp (x) is the minimal polynomial of ii over K. Since an E 0*, we have furthermore that deg(l) = deg(f).

Let now I(x) E o[x] be an arbitrary primitive polynomial, and let

I (x) = fr (x) ... Ir (x)

be its factorization into irreducibles over K. Since 1 = I I I = n I Ii I , mUltiplying the Ii by suitable constants yields I Ii I = 1. The Ii (x) are therefore primitive, irreducible polynomials in o[x]. It follows that

l(x) = 11 (x) ... lr(x) ,

where deg(li) = 0 or deg(li) = deg(fi), and Ii is, up to a constant factor, the power of an irreducible polynomial. If 1 = g h is a factorization into relatively prime polynomials g, h E K[X], then we must have

g = a n Ii , h = b n Ij iEI jEJ

where a,b E K and {I, ... , r} = 1 l:J J and degeJi) = deg(fi) for i E I. We now put

g=anli, h=bn/j, iEI jEJ

for a, b E 0* such that a == a, b == b mod p and I = gh. o

We have introduced henselian fields by a condition of which the reader will find weaker versions in the literature, restricted to monic polynomials only. Both are equivalent as is shown by the following

(6.7) Proposition. A nonarchimedean field (K, v) is henselian if any monic polynomial I(x) E o[x] which splits over the residue class field K = olp as

I(x) == g(x)h (x) mod p

with relatively prime monic factors g (x), h (x) E K [x], admits itself a splitting

I(x) = g(x)h(x)

into monic factors g(x), hex) E o[x] such that

g(x) == g(x) mod p and hex) == hex) mod p.


Proof (E. NART): We have just seen that the property of K to be henselian follows from the condition that the Newton polygon of every irreducible polynomial I(x) = ao + alx + ... + anxn E K[x] is a single line segment. It is therefore sufficient to show this. We may assume that an = 1. Let L I K be the splitting field of I. Then there is always an extension w of v to L. It is obtained for example by taking the completion K of K, extending the

~ "'" valuation of K in a unique way to a valuation v of the algebraic closure K

of K, embedding L into K, and restricting v to L. It is also possible to get the extension w directly, without passing through the completion. For this we refer to [93], chap. XII, §4, tho 1.

Assume now that the Newton polygon of I consists of more than one segment:

m , ,

e: ,

Let the last segment be given by the points (m, e) and (n, 0). If e = 0, we immediately have a contradiction. Because then we have v(ai) ~ 0, so that I(x) E o[x], and ao == ... == am-I == 0 mod p, am ¢ ° mod p. Therefore I(x) == (xn- m + ... + am)Xm mod p, with m > ° because there is more than one segment. In view of the condition of the proposition this contradicts the irreducibility of I.

We will now reduce to e = ° by a transformation. Let ex E L be a root of I(x) of minimum value w(a) and let a E K such that v(a) = e. We consider the characteristic polynomial g(x) of a-Iar E K(a), r = n - m. If I(x) = TI7=1 (x - ai), then g(x) = TI7=1 (x - ara-I). Proposition (6.3) shows that the Newton polygon of g(x) also has more than one segment, the last one of slope

-w(a-1ar ) = v(a) - rw(a) = e - r~ = o. Since g(x) is a power of the minimal polynomial of a-1ar , hence of an irreducible polynomial, this produces the same contradiction as before. D

Let K be a field which is henselian with respect to the exponential valuation v. If L IK is a finite extension of degree n, then v extends uniquely to an exponential valuation w of L, namely

1 w(a) = -V(NLIK(a»).

n


This follows from (6.2) by taking the logarithm. For the value groups and residue class fields of v and w, one gets the inclusions

v(K*) ~ w(L *) and K ~ )...

The index e = e(w I v) = (w(L *) : v(K*))

is called the ramification index of the extension L I K and the degree

f = f(w I v) = [).. : K]

is called the inertia degree. If v, and hence w = ~v 0 NLIK, is discrete and if 0, p, rr, resp. 0, 1.i3, n, are the valuation ring, the maximal ideal and a prime element of K, resp. L, then one has

e = (w(n)Z : v(rr)Z) ,

so that v(rr) = ew(n), and we find

for some unit c: E 0*. From this one deduces the familiar (see chap. I) interpretation of the ramification index: pO = rr CJ = neo = l.i3e, or

(6.8) Proposition. One has [L : K] ::: ef and the fundamental identity

[L : K] = ef,

if v is discrete and L I K is separable.

Proof: Let WI, ... ,WI be representatives of a basis of )..IK and let rro, ... , rre-I E L * be elements the values of which represent the various cosets in w(L *)jv(K*) (the finiteness of e will be a consequence of what follows). If v is discrete, we may choose for instance rri = ni. We show that the elements

Wjrri, j = 1, ... , f, i = 0, ... , e - 1,

are linearly independent over K, and in the discrete case form even a basis of LIK. Let

e-I I L L aijWjrri = 0 i=Oj=I

with aij E K. Assume that not all aij = O. Then there exist nonzero sums

Si = L!=I aijWj, and each time that Si i= 0 we find W(Si) E v(K*). In


fact, dividing Si by the coefficient aiv of minimum value, we get a linear combination of the uy!, ... , wI with coefficients in the valuation ring 0 S; K one of which equals 1. This linear combination is =1= 0 mod s,p, hence a unit, so that w(sj) = w(aiv) E v(K*).

In the sum L~;:ci Si rri, two nonzero summands must have the same value, say W(Sirri) = w(Sjrrj) , i =f. j, because otherwise it could not be zero (observe that w(x) =f. w(y) :::} w(x + y) = min{w(x), w(y)}). It follows that

w(rri) = w(rrj) + w(Sj) - W(Si) == w(rrj) mod v(K*),

a contradiction. This shows the linear independence ofthe Wjrri. In particular, we have ef ~ [L : K].

Assume now that v, and thus also w, is discrete and let n be a prime element in the valuation ring 0 of w. We consider the o-module

e-I I M = L LOWjrri

i=Oj=I

where rri = ni and show that M = 0, i.e., {wjrrd is even an integral basis of 0 over o. We put I

N = L OWj, j=I

so that M = N + nN + ... + ne-IN. We find that

o=N+no,

because, for ex EO, we have ex == atWI + ... + aiwi mod no, ai EO. This implies

0= N + n(N + nO) = ... = N + nN + ... + ne-IN + neo,

so that 0 = M + s,pe = M + pO. Since L IK is separable, 0 is a finitely generated o-module (see chap. I, (2.11)), and we conclude 0 = M from Nakayama's lemma (chap. I, § 11, exercise 7). 0

Remark: We had already proved the identity [L : K] = ef in a somewhat different way in chap. I, (8.2), also in the case where v was discrete and L I K separable. Both hypotheses are actually needed. But, strangely enough, the separability condition can be dropped once K is complete with respect to the discrete valuation. In this case, one deduces the equality 0 = M in the above proof from 0 = M + pO, not by means of Nakayama's lemma, but rather like this: as pi M S; M, we get successively

0= M +p(M +pO) = M +p20 = ... = M +pvO

for all v :::: 1, and since {p v O}vEN is a basis of neighbourhoods of zero in 0, M is dense in O. Since 0 is closed in K, (4.9) implies that M is closed in 0, so that M = O.


Exercise 1. In a henselian field the zeroes of a polynomial are continuous functions of its coefficients. More precisely, one has: let I(x) E K[x] be a monic polynomial of degree n and

r

I(x) = 0 (x - ai)m, i=1

its decomposition into linear factors, with mi :::: I, ai =1= aj for i =1= j. If the monic polynomial g(x) of degree n has all coefficients sufficiently close to those of I(x), then it has r roots f31, ... , f3r which approximate the aI. ... , ar to any previously given precision.

Exercise 2 (Krasner's Lemma). Let a E K be separable over K and let a = aI, ... , an be its conjugates over K. If f3 E K is such that

la - f31 < la - a;! for i = 2, ... , n, then one has K(a) ~ K(f3).

Exercise 3. A field which is henselian with respect to two inequivalent valuations is separably closed (Theorem of F.K. SCHMIDT).

Exercise 4. A separably closed field K is henselian with respect to any nonarchimedean valuation.

More generally, every valuation of K admits a unique extension to any purely inseparable extension L I K .

Hint: If a P = a E K, one is forced to put w(a) = ~v(a). Exercise 5. Let K be a nonarchimedean valued field, v the valuation ring, and p the maximal ideal. K is henselian if and only if every polynomial I(x) = xn + an_IXn- 1 + ... + ao E v[x] such that ao E p and al rI. p has a zero a E p.

Hint: The Newton polygon. Remark: A local ring v with maximal ideal p is called henselian if Hensel's lemma in the sense of (6.7) holds for it. A characterization of these rings which is important in algebraic geometry is the following:

A local ring v is henselian if and only if every finite commutative v-algebra A splits into a direct product A = 0;=1 Ai of local rings Ai.

The proof is not straightforward, we refer to [103], chap. I, §4, tho 4.2.

§ 7. Unramified and Tamely Ramified Extensions

In this section we fix a base field K which is henselian with respect to a nonarchimedean valuation v or 1 I. As before, we denote the valuation ring, the maximal ideal and the residue class field by 0, p, K, respectively. If L 1 K is an algebraic extension, then the corresponding invariants are labelled w, O,~, 1.., respectively. An especially important role among these extensions is played by the unramified extensions, which are defined as follows.

§ 7. Unramified and Tamely Ramified Extensions 153

(7.1) Definition. A finite extension L IK is called unramified if the extension A I K of the residue class field is separable and one has

[L : K] = [A : K].

An arbitrary algebraic extension L I K is called unramified if it is a union of finite unramified subextensions.

Remark: This definition does not require K to be henselian; it applies in all cases where v extends uniquely to L.

(7.2) Proposition. Let L I K and K'I K be two extensions inside an algebraic closure KIK and let L' = LK'. Then one has

LIK unramified ===} L'IK' unramified.

Each subextension of an unramified extension is unramified.

Proof: The notations O,P,K; d,p',K'; O,~,A; O',~',A' are selfexplanatory. We may assume that LIK is finite. Then AIK is also finite and, being separable, is therefore generated by a primitive element ii, A = K(ii). Let a E 0 be a lifting, I(x) E o[x] the minimal polynomial of a and lex) = I(x) mod p E K[X]. Since

[A: K] :s deg(l) = deg(f) = [K(a) : K] :s [L : K] = [A : K],

one has L = K(a) and lex) is the minimal polynomial of ii over K.

We thus have L' = K'(a). In order to prove that L'IK' is unramified, let g(x) E d[x] be the minimal polynomial of a over K' and g(x) = g(x) mod p' E K'[X]. Being a factor of lex), g(x) is separable and hence irreducible over K', because otherwise g(x) is reducible by Hensel's lemma. We obtain

[A' : K'] :s [L' : K'] = deg(g) = deg(g) = [K' (ii) : K'] :s [A' : K'].

This implies [L' : K'] = [A' : K'], i.e., L'IK' is unramified.

If L I K is a subextension of the unramified extension L'I K , then it follows from what we have just proved that L'IL is unramified. Hence so is L IK, by the formula for the degree. 0

(7.3) Corollary. The composite of two unramified extensions of K is again unramified.


Proof: It suffices to show this for two finite extensions LIK and L'IK. LIK is unramified, hence so is LL'IL', by (7.2). This implies that LL'IK is unramified as well because separability is transitive and the degrees of field (and residue field) extensions are mUltiplicative. 0

(7.4) Definition. Let LIK be an algebraic extension. Then the composite T I K of all unramified subextensions is called the maximal unramified subextension of L I K .

(7.5) Proposition. The residue class field of T is the separable closure As of IC in the residue class field extension A I IC of L I K, whereas the value group of T equals that of K .

Proof: Let AD be the residue class field of T and assume ii E A is separable over IC. We have to show that ii E AD. Let l(x) E IC[X] be the minimal polynomial of ii and f(x) E o[x] a monic polynomial such that 1 = f mod p. Then f(x) is irreducible and by Hensel's lemma has a root a in L such that ii = a mod ~, i.e., [K(a) : K] = [IC(ii) : IC]. This implies that K(a)IK is unramified, so that K(a) ~ T, and thus ii E AD.

In order to prove w(T*) = v(K*) we may suppose L IK to be finite. The claim then follows from

[T : K] ~ (w(T*) : v(K*) HAD: IC] = ( w(T*) : v(K*) HT : K]. 0

The composite of all unramified extensions inside the algebraic closure j{ of K is simply called the maximal unramified extension Knr IK of K (nr = 'non ramifiee'). Its residue class field is the separable closure KsIIC. K nr contains all roots of unity of order m not divisible by the characteristic of IC because the separable polynomial xm - 1 splits over K s and hence also over K nr, by Hensel's lemma. If IC is a finite field, then the extension K nr I K is even generated by these roots of unity because they generate K s I IC .

If the characteristic p = char(IC) of the residue class field is positive, then one has the following weaker notion accompanying that of an unramified extension.

(7.6) Definition. An algebraic extension L IK is called tamely ramified if the extension AIIC of the residue class fields is separable and one has ([L : T], p) = 1. In the infinite case this latter condition is taken to mean that the degree of each finite subextension of LIT is prime to p.


As before, in this definition K need not be henselian. We apply it whenever the valuation v of K has a unique extension to L. When the fundamental identity ef = [L : K] holds and AIK is separable, to say that the extension is unramified, resp. tamely ramified, simply amounts to saying that e = 1, resp. (e, p) = 1.

(7.7) Proposition. A finite extension L IK is tamely ramified if and only if the extension LIT is generated by radicals

L = T (m!/iil, ... , mva;-)

such that (mj, p) = 1. In this case the fundamental identity always holds:

[L : K] = ef.

Proof: We may assume that K = T because LIK is obviously tamely ramified if and only if LIT is tamely ramified, and if this is the case, then [T : K] = [A : K] = f. Let LIK be tamely ramified, so that K = A and ([L : K],p) = 1. We first show that e = 1 implies L = K. Let a E L ...... K. Writing a = aI, ... , am for the conjugates and a = Tr(a) = L:7!,1 aj, the element f3 = a - ,ka E L ...... K has trace Tr(f3) = L:r=1 f3j = O. Since v(K*) = w(L *), we may choose abE K* such that v(b) = w(f3) and obtain a unit £ = f3/b E L ...... K with trace L:r=1 £j = O. But the conjugates £j have the same residue classes 8j in A, because A = K. Hence 0 = L:r=1 8j = m8, and thus m == 0 mod p, which contradicts p f [L : K] and ml[L : K].

Now let WI, ••• , Wr E w(L *) be a system of representatives for the quotient w(L *)jv(K*) and mi the order of Wi mod v(K*). Since w(L*) = ~V(NL[K(L*» £; ~v(K*), where n = [L : K], we have mjln, so that (mj,p) = 1. Let Yj E L* be an element such that w(Yj) = Wj. Then W(Yti) = v(Cj), with Cj E K, so that yti = CjE:j for some unit £j in L. As A = K we may write £i = bj Ui, where bj E K and Uj is a unit in L which tends to 1 in A. By Hensel's lemma the equation x mi - Uj = 0 has a solution f3j E L. Putting aj = Yif3j-

1 E L, we find w(aj) = Wi and

m- 1 a j ! = aj , i = , ... , r,

where ai = cjbj E K, i.e., we have K(m~, ... , m:ja;) £; L. By construction, both fields have the same value group and the same residue class field. So, by what we proved first, we have

L = K(m!/iil, ... , ~).

The inequality [L : K] :::s e and thus, in view of (6.8), the equality [L : K] = e, now follows by induction on r. If LI = K(m~, then


WI E w(LD yields

e(LIIK) = (w(LD : v(K*») :::: ml :::: [L I : K].

Also eeL ILl) :::: [L : Ld, because w(L *)/w(Lj) is generated by the residue classes of W2, ... , W r . Thus

e = eeL ILl) e(LIIK) :::: [L : Ld[LI : K] = [L : K].

In order to prove that an extension L = K (m Jfiil, ... , m:;a;) is tamely ramified, it suffices to look at the case r = 1, i.e., L = K ('!!/ii), where (m, p) = 1. The general case then follows by induction. We may assume without loss of generality that K is separably closed. This is seen by passing to the maximal unramified extension KI = K nr , which has the separable closure KI = K s of K as its residue class field. We obtain the following diagram

where L n KI = T = K and LI = KI ('!!/ii). If now LIIKI is tamely ramified, then AIIKI is separable; hence Al = KI and p f [L I : Kd = [L : K] = [L : T], i.e., L IK is also tamely ramified.

Let a = '!!/ii. We may assume that [L : K] = [K('!!/ii) : K] = m. In fact, if d is the greatest divisor of m such that a = a,d for some

m' m' a' E K*, and if m' = mid, then a = Wand [K( W) : K] = m'. Now let n = ord(w(a) mod v(K*». Since mw(a) = v(a) E v(K*), we have m = dn. Consequently wean) = v(b), bE K*, and v(bd ) = w(am ) = v(a); thus am = a = sbd for some unit s in K. As (d, p) = 1, the equation x d - s = 0 splits over the separably closed residue field K into distinct linear factors, hence also over K by Hensel's lemma. Therefore am = bd = a for some new b E K*. Since xm - a is irreducible, we have d = 1, and hence m = n. Thus

e :::: n = [L : K] :::: ef :::: e,

in other words f = 1, and so A = K and p t n = e. This shows that LIK is tamely ramified. 0

(7.8) Corollary. Let L I K and K 'I K be two extensions inside the algebraic closure KIK, and L' = LK'. Then we have:

L I K tamely ramified =} L 'I K' tamely ramified.

Every subextension of a tamely ramified extension is tamely ramified.

§7. Unramified and Tamely Ramified Extensions 157

Proof: We may assume without loss of generality that L IK is finite and consider the diagram

L--L' I I T--T' I I K--K'.

The inclusion T ~ T' follows from (7.2). If L IK is tamely ramified, then L = T(m,;;al, ... , m:;a;), (mj,p) = 1; hence L' = LK' = LT' = T,(m,;;al, ... , m:;a;), so that L'IK' is also tamely ramified, by (7.7).

The claim concerning the subextensions follows exactly as in the unramified case. 0

(7.9) Corollary. The composite of tamely ramified extensions is tamely ramified.

Proof: This follows from (7.8), exactly as (7.3) followed from (7.2) in the unramified case. 0

(7.10) Definition. Let LIK be an algebraic extension. Then the composite V I K of all tamely ramified subextensions is called the maximal tamely ramified subextension of L I K .

Let w(L *)(p) denote the subgroup of all elements Q) E w(L *) such that mQ) E v(K*) for some m satisfying (m, p) = 1. The quotient group w(L *)(p) jv(K*) then consists of all elements of w(L *)jv(K*) whose order is prime to p.

(7.11) Proposition. The maximal tamely ramified subextension V I K of L I K has value group w(V*) = w(L *)(p) and residue class field equal to the separable closure As ofK in AIK.

Proof: We may restrict to the case of a finite extension L/K. By passing from K to the maximal unramified subextension, we may assume by (7.5) that As = K. As P t e(V/K) = #w(V*)jv(K*), we certainly have w(V*) ~ w(L *)(p). Conversely we find, as in the proof of (7.7), for every Q) E w(L *)(p) a radical a = Va E L such that a E K, (m, p) = 1 and w(a) = Q), so that one has a E V, and Q) E w(V*). 0


The results obtained in this section may be summarized in the following picture:

K c T c V c L

K C As = As c A

v(K*) w(T*) c w(L *)(p) c w(L *).

If LIK is finite and e = e'pQ where (e',p) = 1, then [V: T] = e'. The extension L IK is called totally (or purely) ramified if T = K, and wildly ramified if it is not tamely ramified, i.e., if V :j:. L.

Important Example: Consider the extension Qp({)IQp for a primitive n -th root of unity 1;. In the two cases (n, p) = 1 and n = pS, this extension behaves completely differently. Let us first look at the case (n, p) = 1 and choose as our base field, instead of Qp, any discretely valued complete field K with finite residue class field K = IF' q' with q = pr.

(7.12) Proposition. Let L = K(I;), and letOlo, resp. AIK, be the extension of valuation rings, resp. residue class fields, of L I K. Suppose that (n, p) = 1. Then one has:

(i) The extension L I K is unramified of degree f, where f is the smallest natural number such that qf == 1 mod n.

(ii) The Galois group G(LIK) is canonically isomorphic to G(AIK) and is generated by the automorphism <p : I; 1-+ I;q.

(iii) 0 = o[l;].

Proof: (i) If if>(X) is the minimal polynomial of I; over K, then the reduction <iJ (X) is the minimal polynomial of f = I; mod ~ over K.

Indeed, being a divisor of xn - 1, <iJ(X) is separable and by Hensel's lemma cannot split into factors. if> and <iJ have the same degree, so that [L : K] = [K(f) : K] = [A : K] =: f. L IK is therefore unramified. The polynomial xn - 1 splits over 0 and thus (because (n, p) = 1) over A into distinct linear factors, so that A = IF' q! contains the group J.Ln of n-th roots of unity and is generated by it. Consequently f is the smallest number such that J.Ln S; IF';!, i.e., such that n I ql - 1. This shows (i). (ii) results trivially

from this. (iii) Since LIK is unramified, we have pO =~, and since 1,1;, ... , 1;1- 1

represents a basis of AIK, we have 0 = o[l;] + pO, and 0 = o[l;] by Nakayama's lemma. 0


(7.13) Proposition. Let ~ be a primitive pm -th root of unity. Then one has:

(i) Qp(t)IQp is totally ramified of degree cp(pm) = (p - 1)pm-l.

(ii) G(Qp(S)IQp) ~ (71,/pm71,)*.

(iii) 71,p[n is the valuation ring ofQp(S).

(iv) 1 - ~ is a prime element of71,p[n with norm p.

Proof: ~ = ~pm-l is a primitive p-th root of unity, i.e.,

~P-l + ~p-2 + ... + 1 = 0, hence

~(p_l)pm-l + ~(p_2)pm-l + ... + 1 = 0.

Denoting by </Y the polynomial on the left, ~ - 1 is a root of the equation </Y (X + 1) = 0. But this is irreducible because it satisfies Eisenstein's criterion: </YO) = p and </y(X) == (Xpm - 1)/(Xpm-l - 1) = (X - l)pm-l(p_1) mod p.

It follows that [Qp(t) : Qpl = cp(pm). The canonical injection G(Qp(S) IQp) -+ (71,/ pm71,)*, a 1--+ n(a), where a~ = ~n((J), is therefore bijective, since both groups have order cp(pm). Thus

MQlp(;)IQp 0 - S) = TI 0 - at) = </YO) = p. (J

Writing w for the extension of the normalized valuation vp of Qp' we find furthermore that cp(pm)w(~ - 1) = vp(p) = 1, i.e., Qp(t)IQp is totally ramified and ~ - 1 is a prime element of Qp(t). As in the proof of (6.8), it follows that 71,p[~ - 1] = 71,p[n is the valuation ring of Qp(t). This concludes the proof. 0

If ~n is a primitive n -th root of unity and n = n' pm, with (n', p) = 1, then propositions (7.12) and (7.13) yield the following result for the maximal unramified and the maximal tamely ramified extension:

Exercise 1. The maximal unramified extension of Qlp is obtained by adjoining all roots of unity of order prime to p.

Exercise 2. Let K be henselian and K nr I K the maximal unramified extension. Show that the subextensions of K nr I K correspond 1-1 to the subextensions of the separable closure Ks IK.

Exercise 3. Let L I K be totally and tamely ramified, and let L1, resp. r, be the value group of L, resp. K. Show that the intermediate fields of L I K correspond 1-1 to the subgroups of L1 / r.


§ 8. Extensions of Valuations

Having seen that the henselian valuations extend uniquely to algebraic extensions we will now study the question of how a valuation v of a field K extends to an algebraic extension in general. So let v be an arbitrary archimedean or nonarchimedean valuation. There is a little discrepancy in notation here, because archimedean valuations manifest themselves only as absolute values while the letter v has hitherto been used for nonarchimedean exponential valuations. In spite of this, it will prove advantageous, and agrees with current usage, to employ the letter v simultaneously for both types of valuations, to denote the corresponding multiplicative valuation in both cases by I Iv and the completion by Kv. Where confusion lurks, we will supply clarifying remarks.

For every valuation v of K we consider the completion K v and an algebraic closure K v of K v. The canonical extension of v to K v is again denoted by v and the unique extension of this latter valuation to K v by v.

Let L I K be an algebraic extension. Choosing a K -embedding

we obtain by restriction of v to r L an extension

w = var

of the valuation v to L. In other words, if v, resp. V, are given by the absolute values I I v, resp. I I v' on K, K v, resp. K v, where I I v extends precisely the absolute value I I v of K v, then we obtain on L the multiplicative valuation

Ixlw = Irxlv·

The mapping r : L -+ K v is obviously continuous with respect to this valuation. It extends in a unique way to a continuous K -embedding

where, in the case of an infinite extension L IK, Lw does not mean the completion of L with respect to w, but the union Lw = Ui Liw of the completions Liw of all finite subextensions Li IK of LIK. This union will be henceforth called the localization of L with respect to w. When [L : K] < 00, r is given by the rule

x = w -lim Xn f-----+ rx:= v -lim rxn , n---+oo n---+oo

§ 8. Extensions of Valuations 161

where {Xn}nEN is a w-Cauchy sequence in L, and hence {rxn}nEN a v -Cauchy sequence in Kv. Note here that the sequence rXn converges in the finite complete extension r L . K v of K v. We consider the diagram of fields

L--Lw

(*) I I

K

________ Kv

The canonical extension of the valuation w from L to Lw is precisely the unique extension of the valuation v from K v to the extension L w I K v' We have

Lw = LKv ,

because if LIK is finite, then the field LKv ~ Lw is complete by (4.8), contains the field L and therefore has to be its completion. If L w I K v has degree n < 00, then, by (4.8), the absolute values corresponding to v and w satisfy the relation

Ixlw = 1INLwIKv(x)lv'

The field diagram (*) is of central importance for algebraic number theory. It shows the passage from the "global extension" L I K to the "local extension" L w I K v and thus represents one of the most important methods of algebraic number theory, the so-called local-to-global principle. This terminology arises from the case of a function field K, for example K = e(t), where the elements of the extension L are algebraic functions on a Riemann surface, hence on a global object, whereas passing to Kv and Lw signifies looking at power series expansions, i.e., the local study of functions. The diagram (*)

thus expresses in an abstract manner our original goal, to provide methods of function theory for use in the theory of numbers by means of valuations.

We saw that every K -embedding r : L -+ K v gave us an extension w = vo r of v. For every automorphism (j E G(KvIKv) of Kv over Kv, we obtain with the composite

T - a -L --+ Kv --+ Kv

a new K -embedding r' = (j 0 r of L. It will be said to be conjugate to r over K v. The following result gives us a complete description of the possible extensions of v to L.

(8.1) Extension Theorem. Let L IK be an algebraic field extension and v a valuation of K. Then one has:


(i) Every extension w of the valuation v arises as the composite w = v 0 r for some K -embedding r : L ~ K v.

(ii) Two extensions v 0 r and vor' are equal if and only if r and r' are conjugate over K v.

Proof: (i) Let w be an extension of v to L and Lw the localization of the canonical valuation, which is again denoted by w. This is the unique extension of the valuation v from Kv to Lw. Choosing any Kv-embedding r : Lw ~ Kv , the valuation v 0 r has to coincide with w. The restriction of r to L is therefore a K -embedding r : L ~ K v such that w = v 0 r.

(ii) Let r and a or, with a E G(KvIKv), be two embeddings of L conjugate over K v. Since v is the only extension of the valuation v from K v to K v, one has v = VO a, and thus v 0 r = v 0 (a 0 r). The extensions induced to L by r and by a 0 r are therefore the same.

Conversely, let r, r' : L ~ K v be two K -embeddings such that VO r = VO r/. Let a : rL ~ r'L be the K-isomorphism a = r'o r-1.

We can extend a to a Kv-isomorphism

a: rL· Kv ~ r'L· Kv.

Indeed, r L is dense in r L . K v, so every element x E r L . K v can be written as a limit

x = lim rXn n~oo

for some sequence Xn which belongs to a finite subextension of L. As v 0 r = vor', the sequence r'xn = arxn converges to an element

ax := lim arxn n~oo

in r' L . K v. Clearly the correspondence x 1--+ a x does not depend on the choice of a sequence {xn }, and yields an isomorphism rL . Kv ~ r'L . Kv which leaves Kv fixed. Extending a to a Kv-automorphism ij E G(KvIKv) gives r' = ij 0 r, so that r and r' are indeed conjugate over Kv. 0

Those who prefer to be given an extension L I K by an algebraic equation I (X) = 0 will appreciate the following concrete variant of the above extension theorem.

Let L = K (ex) be generated by the zero ex of an irreducible polynomial I(X) E K[X] and let


be the decomposition of I(X) into irreducible factors II (X), ... , Ir(X) over the completion Kv. Of course, the mi are one if I is separable. The Kembeddings T : L -+ K v are then given by the zeroes {3 of I (X) which lie in Kv:

T: L ---+ K v , T(a) = {3.

Two embeddings T and T' are conjugate over Kv if and only ifthe zeroes T(a) and T' (a) are conjugate over K v , i.e., ifthey are zeroes ofthe same irreducible factor if. With (8.1), this gives the

(8.2) Proposition. Suppose the extension L I K is generated by the zero a of the irreducible polynomial I (X) E K [X].

Then the valuations WI, •.. , Wr extending v to L correspond 1-1 to the irreducible factors II, ... , Ir in the decomposition

of lover the completion K v.

The extended valuation Wi is explicitly obtained from the factor Ii as follows: let ai E Kv be a zero of Ii and let

Ti : L -+ K v , a f-+ ai,

be the corresponding K -embedding of L into K v. Then one has

Wi = VO Ti.

Ti extends to an isomorphism

Ti : LWi ~ Kv(ai)

on the completion LWi of L with respect to Wi.

Let L I K be again an arbitrary finite extension. We will write W I v to indicate that W is an extension of the valuation v of K to L. The inclusions L "-+ L W induce homomorphisms L ® K K v -+ L W via a ® b f-+ ab, and hence a canonical homomorphism

({J : L ®K Kv ---+ fl Lw. wlv

To begin with, the tensor product is taken in the sense of vector spaces, i.e., the K -vector space L is lifted to a Kv-vector space L®K Kv. This latter, however, is in fact a K v -algebra, with the multiplication (a ® b) (a' ® b') = aa' ® bb' , and ({J is a homomorphism of K v -algebras. This homomorphism is the subject of the


(8.3) Proposition. If L IK is separable, then L ®K Kv ~ TIwlv Lw.

Proof: Let a be a primitive element for L I K, so that L = K (a), and let I(X) E K[X] be its minimal polynomial. To every wlv, there corresponds an irreducible factor Iw(X) E Kv[X] of I(X), and in view of the separability, we have I(X) = TIwlv Iw(X). Consider all the Lw as embedded into an

algebraic closure Kv of Kv and denote by aw the image of a under L ---+ Lw. Then we find Lw = Kv(aw) and Iw(X) is the minimal polynomial of aw over Kv. We now get a commutative diagram

KdXJ/(f) -----+) TI Kv[X]/(fw)

1 wlv 1 TIL w , wlv

where the top arrow is an isomorphism by the Chinese remainder theorem. The arrow on the left is induced by X H- a ® I and is an isomorphism because K[X]/(f) ~ K(a) = L. The arrow on the right is induced by X H- aw and is an isomorphism because Kv[X]/(fw) - Kv(aw) = Lw. Hence the bottom arrow is an isomorphism as well. 0

(8.4) Corollary. If L IK is separable, then one has

and

[L : K] = I)Lw : Kv]

NLIK(a) = TI NLwlKuCa ) , wlv

wlv

TrLIK(a) = L TrLwIKv(a). wlv

Proof: The first equation results from (8.3) since [L : K] = dimK(L) = dimKv(L ®K Kv). On both sides of the isomorphism

L ®K Kv ~ TI Lw wlv

let us consider the endomorphism: multiplication by a. The characteristic polynomial of a on the K v -vector space L ® K K v is the same as that on the K -vector space L. Therefore

char. polynomialL1K(a) = TI char. polynomialLwIKv(a). wlv

This implies immediately the identities for the norm and the trace. 0


If v is a nonarchimedean valuation, then we define, as in the henselian case, the ramification index of an extension w I v by

ew = ( w(L *) : v(K*))

and the inertia degree by

fw = p,w : K],

where Aw, resp. K, is the residue class field of w, resp. v. From (8.4) and (6.8), we obtain the fundamental identity of valuation theory:

(8.5) Proposition. If v is discrete and L I K separable, then

Lewfw=[L:K]. wlv

This proposition repeats what we have already seen in chap. I, (8.2), working with the prime decomposition. If K is the field of fractions of a Dedekind domain v, then to every nonzero prime ideal p of v is associated the p-adic valuation vp of K, defined by vp(a) = \!p, where (a) = TIp pVp (see chap. I, § 11, p. 67). The valuation ring of vp is the localization vp. If 0 is the integral closure of v in L and if

pO = ~~1 •• • ~~r

is the prime decomposition of p in L, then the valuations Wi = t V\jJi'

i = 1, ... , r, are precisely the extensions of v = vp to L, ei are the corresponding ramification indices and Ii = [O/~i : vip] the inertia degrees. The fundamental identity

r

Ledi = [L: K] i=l

has thus been established in two different ways. The raison d' etre of valuation theory, however, is not to reformulate ideal-theoretic knowledge, but rather, as has been stressed earlier, to provide the possibility of passing from the extension L I K to the various completions L w I K v where much simpler arithmetic laws apply. Let us also emphasize once more that completions may always be replaced with henselizations.

Exercise 1. Up to equivalence, the valuations of the field QCJS) are given as follows.

1) la + bJ511 = la + bJ51 and la + bJ512 = la - bJ51 are the archimedean valuations.


2) If p = 2 or 5 or a prime number i= 2, 5 such that ( ~) = -1, then there is

exactly one extension of I Ip to 1Q(.J5), namely

la + b.J5lp = la2 - 5b21!/2.

3) If p is a prime number i= 2,5 such that (~) = 1, then there are two

extensions of I Ip to 1Q(.J5), namely

la+b.J5I P1 = la+bylp, resp. la+b.J5l p2 = la-bylp, where y is a solution of x 2 - 5 = 0 in IQp.

Exercise 2. Determine the valuations of the field lQ(i) of the Gaussian numbers.

Exercise 3. How many extensions to IQ (-V2) does the archimedean absolute value I of IQ admit?

Exercise 4. Let L I K be a finite separable extension, C) the valuation ring of a discrete valuation v and 0 its integral closure in L. If w I v varies over the extensions of v to L and ov, resp. Ow, are the valuation rings of the completions Kv, resp. L w, then one has

Exercise 5. How does proposition (8.2) relate to Dedekind's proposition, chap. I, (8.3)?

Exercise 6. Let L I K be a finite field extension, v a nonarchimedean exponential valuation, and w an extension to L. If 0 is the integral closure of the valuation ring C)

of v in L, then the localization 0'.l3 of 0 at the prime ideal ~ = {ex EO I w(ex) > O} is the valuation ring of w.

§ 9. Galois Theory of Valuations

We now consider Galois extensions L I K and study the effect of the Galois action on the extended valuations w I v. This leads to a direct generalization of "Hilbert's ramification theory" - see chap. I, §9, where we studied, instead of valuations v, the prime ideals p and their decomposition p = ~11 ... ~~r in Galois extensions of algebraic number fields. The arguments stay the same, so we may be rather brief here. However, we formulate and prove all results for extensions that are not necessarily finite, using infinite Galois theory. The reader who happens not to know this theory should feel free to assume all extensions in this section to be finite. On the other hand, we treat infinite Galois theory also in chap. N, § 1 below. Its main result can be put in a nutshell like this:

In the case of a Galois extension L I K of infinite degree, the main theorem of ordinary Galois theory, concerning the 1-1 correspondence between

§ 9. Galois Theory of Valuations 167

the intermediate fields of L I K and the subgroups of the Galois group G(LIK) ceases to hold; there are more subgroups than intermediate fields. The correspondence can be salvaged, however, by considering a canonical topology on the group G(L IK), the Krull topology. It is given by defining, for every a E G(L IK), as a basis of neighbourhoods the cosets aG(L 1M), where M I K varies over the finite Galois subextensions of L I K. G (L I K) is thus turned into a compact, Hausdorff topological group. The main theorem of Galois theory then has to be modified in the infinite case by the condition that the intermediate fields of L I K correspond 1-1 to the closed subgroups of G(L IK). Otherwise, everything goes through as in the finite case. So one tacitly restricts attention to closed subgroups, and accordingly to continuous homomorphisms of G(L IK).

SO let L I K be an arbitrary, finite or infinite, Galois extension with Galois group G = G(L IK). If v is an (archimedean or nonarchimedean) valuation of K and w an extension to L, then, for every a E G, w 0 a also extends v, so that the group G acts on the set of extensions w I v.

(9.1) Proposition. The group G acts transitively on the set of extensions wlv, i.e., every two extensions are conjugate.

Proof: Let w and Wi be two extensions of v to L. Suppose L IK is finite. If wand Wi are not conjugate, then the sets

{ w 0 a I a E G} and {Wi 0 a I a E G}

would be disjoint. By the approximation theorem (3.4), we would be able to find an x E L such that

laxl w < 1 and laxl wl > 1

for all a E G. Then one would have for the norm ot = NLIK(X) = TIcrEG ax that loti v = TIcr I a x I w < 1 and likewise loti v > 1, a contradiction.

If L I K is infinite, then we let M I K vary over all finite Galois subextensions and consider the sets X M = {a E G I w 0 a 1M = Wi 1M}. They are nonempty, as we have just seen, and also closed because, for a E G " X M ,

the whole open neighbourhood aG(L 1M) lies in the complement of XM. We have nM XM #- 0, because otherwise the compactness of G would yield a relation n~=l XMi = 0 with finitely many Mj, and this is a contradiction because if M = MI'" Mr , then XM = n~=l XMi • 0

(9.2) Definition. The decomposition group of an extension w of v to L is defined by

Gw = Gw(LIK) = {a E G(LIK) I w 0 a = w} .


If v is a nonarchimedean valuation, then the decomposition group contains two further canonical subgroups

G w ;2 lw ;2 Rw ,

which are defined as follows. Let a, resp. a, be the valuation ring, p, resp. If>, the maximal ideal, and let K = alp, resp. A = a/If>, be the residue class field of v, resp. w.

(9.3) Definition. The inertia group of w I v is defined by

lw = Iw(LIK) = {a E G w I ax:: X mod If> for all x E a} and the ramification group by

Rw=Rw(LIK)={aEGwl ([XX ::lmodlf> fora11 XEL*}.

Observe in this definition that, for a E G w, the identity w 0 a = w implies that one always has a a = a and a x/x E a, for all x E L * .

The subgroups G w, lw, Rw of G = G(L IK), and in fact all canonical subgroups we will encounter in the sequel, are all closed in the Krull topology. The proof of this is routine in all cases. Let us just illustrate the model of the argument for the example of the decomposition group.

Let a E G = G(L IK) be an element which belongs to the closure of G w. This means that, in every neighbourhood aG(L 1M), there is some element aM of G w. Here MIK varies over all finite Galois subextensions of LIK. Since aM E aG(LIM), we have aMIM = aiM, and w oaM = w implies that w 0 a 1M = W 0 aM 1M = wi M. As L is the union of all the M, we get w 0 a = w, so that a E G w • This shows that the subgroup G w is closed in G.

The groups G w, lw, Rw carry very significant information about the behaviour of the valuation v of K as it is extended to L. But before going into this, we will treat the functorial properties of the groups G w, lw, Rw.

Consider two Galois extensions L IK and L'IK' and a commutative diagram

L T) L'

T T K~K'


with homomorphisms r which will typically be inclusions. They induce a homomorphism

r*: G(L'IK') ~ G(LIK), r*(a') = r-1a'r.

Observe here that, L I K being normal, the same is true of r L IrK, and thus one has a'rL ~ rL, so that composing with r- 1 makes sense.

Now let w' be a valuation of L', v' = W'IK' and w = w' 0 r, v = WIK. Then we have the

(9.4) Proposition. r*: G(L'IK') -+ G(LIK) induces homomorphisms

Gw,(L'IK') ~ Gw(LIK),

Iw,(L'IK') ~ Iw (LIK),

Rw,(L'IK') ~ Rw(LIK). In the latter two cases, v is assumed to be nonarchimedean.

Proof: Let a' E Gw,(L'IK') and a = r*(a'). If x E L, then one has

Ixl wOCT = laxl w = Ir- 1a'rxl w = la'rxl w' = Irxl w' = Ixl w,

so that a E Gw(L IK). If a' E Iw,(L'IK') and x EO, then

w(ax - x) = w( r-\a'rx - rx)) = w'( a'(rx) - (rx)) > 0,

and a E Iw(L IK). If a' E Rw,(L'IK') and x E L *, then

w(axx -1) = w(r-1(a~:x -1)) = w,(a~:x -1) > 0, so that a E Rw(LIK). 0

If the two homomorphisms r : L -+ L' and r : K -+ K' are isomorphisms, then the homomorphisms (9.4) are of course isomorphisms. In particular, in the case K = K', L = L', we find for each r E G(LIK):

Gwor = r-1G w r, Iwor = r-1 Iwr, Rwor = r-1 Rwr ,

i.e., the decomposition, inertia, and ramification groups of conjugate valuations are conjugate.

Another special case arises from an intermediate field M of L I K by the diagram

L L

K ) M.

r* then becomes the inclusion G(L 1M) y. G(L IK), and we trivially get the


(9.5) Proposition. For the extensions K ~ M ~ L, one has

Gw(LIM) Gw(LIK)nG(LIM),

Iw(LIM) = Iw(LIK) n G(LIM),

Rw(LIM) = Rw(LIK) n G(LIM).

A particularly important special case of (9.4) occurs with the diagram

L_Lw

I I ______ Kv

K

which can be associated to any extension of valuations w I v of L I K . If L I K is infinite, then Lw has to be read as the localization in the sense of §8, p. 160. (This distinction is rendered superfluous if we consider, as we may perfectly well do, the henselization of LIK.) Since in the local extension LwlKv the extension of the valuation is unique, we denote the decomposition, inertia, and ramification groups simply by G(Lw IKv), I (Lw IKv), R(Lw IKv). In this case, the homomorphism r* is the restriction map

G(Lw IKv) ----* G(L IK), (j f----* (j IL,

and we have the

(9.6) Proposition. Gw(LIK) ~ G(LwIKv),

Iw(LIK) ~ I(LwIKv),

Rw(LIK) ~ R(LwIKv)·

Proof: The proposition derives from the fact that the decomposition group Gw(LIK) consists precisely of those automorphisms (j E G(LIK) which are continuous with respect to the valuation w. Indeed, the continuity of the (j E Gw(L IK) is clear. For an arbitrary continuous automorphism (j, one has

Ixl w < 1 ===} l(jxl w = IxlwQO' < 1,

because Ix I w < 1 means that xn and hence also (j xn is a w -nullsequence, i.e., l(jxl w < 1. By §3, p.1l7, this implies that wand w O(j are equivalent, and hence in fact equal because WIK = W 0 (jIK, so that (j E Gw(LIK).

Since L is dense in L w, every (j E Gw(LIK) extends uniquely to a continuous K v -automorphism {j of L w and it is clear that {j E I (L wi K v), resp. (j E R(LwIKv), if (j E Iw(LIK), resp. (j E Rw(LIK). This proves the bijectivity of the mappings in question in all three cases. 0


The above proposition reduces the problems concerning a single valuation of K to the local situation. We identify the decomposition group Gw(L IK) with the Galois group of L w I K v and write

Gw(LIK) = G(LwIKv),

and similarly Iw(LIK) = I(LwIKv) and Rw(LIK) = R(LwIKv).

We now explain the concrete meaning of the subgroups G w , I w , Rw of G = G(LIK) for the field extension LIK.

The decomposition group G w consists - as was shown in the proof of (9.6) - of all automorphisms a E G that are continuous with respect to the valuation w. It controls the extension of v to L in a group-theoretic manner. Denoting by Gw \G the set of all right cosets Gwa, by Wv the set of extensions of v to L and choosing a fixed extension w, we obtain a bijection

Gw\G ~ Wv , Gwa f--* wa.

In particular, the number #Wv of extensions equals the index (G : G w ). As mentioned already in chap. I, § 9 - and left for the reader to verify - the decomposition group also describes the way a valuation v extends to an arbitrary separable extension L I K. For this, we embed L I K into a Galois extension NIK, choose an extension w of v to N, and put G = G(NIK), H = G(NIL), G w = Gw(NIK), to get a bijection

Gw\G/H ~ Wv , GwaH f--* woalL.

(9.7) Definition. The fixed field of G w ,

Z w = Z w (L I K) = {x ELI a x = x for all a E G w } ,

is called the decomposition field of w over K.

The rOle of the decomposition field in the extension L I K is described by the following proposition.

(9.8) Proposition.

(i) The restriction Wz of w to Zw extends uniquely to L.

(ii) If v is nonarchimedean, Wz has the same residue class field and the same value group as v.

(iii) Zw = L n Kv (the intersection is taken inside Lw).


Proof: (i) An arbitrary extension w' of Wz to L is conjugate to w over Zw; thus w' = w 0 (J', for some (J' E G(L I Zw) = Gw, i.e., w' = w.

(iii) The identity Zw = L n Kv follows immediately from Gw(LIK) ;;:: G(LwIKv).

(ii) Since K v has the same residue class field and the same value group as K, the same holds true for Zw = L n Kv. 0

The inertia group I w is defined only if w is a nonarchimedean valuation of L. It is the kernel of a canonical homomorphism of Gw . For if 0 is the valuation ring of w and \13 the maximal ideal, then, since (J'O = 0 and (J'\13 =~, every (J' E G w induces a K-automorphism

(f : 0/\13 ---+ 0/\13, x mod \13 1---+ (J' x mod \13,

of the residue class field A, and we obtain a homomorphism

with kernel I w.

(9.9) Proposition. The residue class field extension A IK is normal, and we have an exact sequence

Proof: In the case of a finite Galois extension, we have proved this already in chap. I, (9.4). In the infinite case AIK is normal since L IK, and hence also A I K, is the union of the finite normal subextensions. In order to prove the surjectivity of I : Gw --+ G(AIK) all one has to show is that I(G w) is dense in G(AIK) because I(G w ), being the continuous image of a compact set, is compact and hence closed. Let (Y E G(AIK) and (yG(AIJL) be a neighbourhood of (Y, where JLIK is a finite Galois subextension of AIK. We have to show that this neighbourhood contains an element of the image I (r), r E Gw. Since Zw has the residue class field K, there exists a finite Galois subextension M I Zw of L I Zw whose residue class field M contains the field JL. As G(MIZw) --+ G(MIK) is surjective, the composite

Gw = G(LIZw) ---+ G(MIZw) ---+ G(MIK) ---+ G(JLIK)

is also surjective, and if r E Gw is mapped to (YIIL' then I(r) E (yG(AIJL), as required. 0


(9.10) Definition. The fixed field of I w ,

Tw = Tw(LIK) = {x ELI (jX =X for all (j E Iw},

is called the inertia field of w over K.

For the inertia field, (9.9) gives us the isomorphism

It has the following significance for the extension L I K .

(9.11) Proposition. T wi Zw is the maximal unramified subextension of L I Zw.

Proof: By (9.6), we may assume that K = Zw is henselian. Let T IK be the maximal unramified subextension of L I K. It is Galois, since the conjugate extensions are also unramified. By (7.5), T has the residue class field As, and we have an isomorphism

G(TIK) ~ G(AsIK).

Swjectivity follows from (9.9) and the injectivity from the fact that T IK is unramified: every finite Galois subextension has the same degree as its residue class field extension. An element (j E G(L IK) therefore induces the identity on As, i.e., on A, if and only if it belongs to G(L IT). Consequently, G(LIT) = I w, hence T = Tw. 0

If in particular K is a henselian field and Ks IK its separable closure, then the inertia field of this extension is the maximal unramified extension T I K and has the separable closure K s IK as its residue class field. The isomorphism

shows that the unramified extensions of K correspond 1-1 to the separable extensions of K.

Like the inertia group, the ramification group Rw is the kernel of a canonical homomorphism

Iw ---+ x(LIK),

where x(LIK) = Hom(L1jr,A*),


where ..1 = w(L *), and r = v(K*). If a E Iw, then the associated homomorphism

Xu : ..1/ r -+ A *

is given as follows: for 8 = 8 mod r E ..1/ r, choose an x E L * such that w(x) = 8 and put

- ax Xu (8) = - mod >,p.

x

This definition is independent of the choice of the representative 8 E 8 and of x E L *. For if x' E L * is an element such that w (x') == w (x) mod r, then w(x') = w(xa), a E K*. Then x' = xau, u E 0*, and since au/u == 1 mod>,p (because a E /r;f3), one gets ax'/x' == ax/x mod >,p.

One sees immediately that mapping a 1-+ Xu is a homomorphism Iw -+ x(LIK) with kernel Rw.

(9.12) Proposition. Rw is the unique p-Sylow subgroup of Iw.

Remark: If L I K is a finite extension, then it is clear what this means. In the infinite case it has to be understood in the sense of profinite groups, i.e., all finite quotient groups of Rw, resp. Iw/ Rw, by closed normal subgroups have p-power order, resp. an order prime to p. In order to understand this better, we refer the reader to chap. IV, §2, exercise 3-5.

Proof of (9.12): By (9.6), we may assume that K is henselian. We restrict to the case where L I K is a finite extension. The infinite case of the proposition follows formally from this.

If Rw were not a p-group, then we would find an element a E Rw of prime order £ =1= p. Let K' be the fixed field of a and K' its residue class field. We show that K' = A. Since Rw ~ Iw, we have that T ~ K'. Thus As ~ K', so that A IK' is purely inseparable and of p-power degree. On the other hand, the degree has to be a power of £, for if ii E A and if a E L is a lifting of ii, and f (x) E K' [x] is the minimal polynomial of a over K', then lex) = g(x)m, where g(x) E K'[X] is the minimal polynomial of ii over K', which has degree either 1 or £, as this is so for f(x). Thus we have indeed K' = A, so that L IK' is tamely ramified, and by (7.7) is of the form L = K'(a) with a = !ta, a E K'. It follows that aa = ta, with a primitive £-th root of unity t E K'. Since a E Rw, we have on the other hand aa/a = t == 1 mod >,p, a contradiction. This proves that Rw is a p-group.

Since p = char(A), the elements in A * have order prime to p, provided they are of finite order. The group X (L I K) = Hom(L1/ r, A *) therefore has order prime to p. This also applies to the group Iw/ Rw ~ X (L IK), so that Rw is indeed the unique p-Sylow subgroup. 0

§ 9. Galois Theory of Valuations

(9.13) Definition. The fixed field of Rw,

Vw = Vw(LIK) = {x ELI ax = x forall a E Rw},

is called the ramification field of w over K.

175

(9.14) Proposition. V wi Zw is the maximal tamely ramified subextension ofLIZw.

Proof: By (9.6) and the fact that the value group and residue class field do not change, we may assume that K = Zw is henselian. Let Vw be the fixed field of Rw. Since Rw is the p-Sylow subgroup of I w, Vw is the union of all finite Galois subextensions of LIT of degree prime to p. Therefore V w contains the maximal tamely ramified extension V of T (and thus of Zw). Since the degree of each finite subextension M I V of V w I V is not divisible by p, the residue field extension of MJV is separable (see the argument in the proof of (9.12». Therefore V w I V is tamely ramified, and V w = V. D

(9.15) Corollary. We have the exact sequence

1 -+ Rw -+ Iw -+ x(LIK) -+ 1.

Proof: By (9.6) we may assume, as we have already done several times before, that K is henselian. We restrict to considering the case of a finite extension L IK. In the infinite case the proof follows as in (9.9). We have already seen that Rw is the kernel of the arrow on the right. It therefore suffices to show that

As T w I K is the maximal unramified subextension of V w I K, V wiT w has inertia degree 1. Thus, by (7.7),

[V w : T w] = #( w(V~) jw(T~») .

Furthermore, by (7.5), we have w(T~) = v(K*) =: r, and putting Ll = w(L *), we see that w(V~)jv(K*) is the subgroup Ll(p) j r of Llj r consisting of all elements of order prime to p, where p = char(K). Thus

[Vw: Tw] =#(Ll(p)jr).

Since}.. * has no elements of order divisible by p, we have on the other hand that

x(LIK) = Hom(Lljr,}..*) = Hom(Ll(p) /r,}..*).


But C7.7) implies that J.. * contains the m-th roots of unity whenever ,tj,(p) / r contains an element of order m, because then there is a Galois extension generated by radicals TwCVa)ITw of degree m. This shows that xCLIK) is the Pontryagin dual of the group L1 (p) / r so that indeed

o

Exercise 1. Let K be a henselian field, L I K a tamely ramified Galois extension, G = G(LIK), 1= I(LIK) and r = Gil = G(J..IK). Then I is abelian and becomes a r -module by letting u = u I E r operate on I via 'l' 1-+ U'l'U-1•

Show that there is a canonical isomorphism I ~ x(LIK) of r-modules. Show furthermore that every tamely ramified extension can be embedded into a tamely ramified extension LIK, such that G is the semi-direct product of x(LIK) with G().,IK): G ~ x(LIK) >4 G().,IK).

Hint: Use (7.7).

Exercise 2. The maximal tamely ramified abelian extension V of Qp is finite over the maximal unramified abelian extension T of Qp.

Exercise 3. Show that the maximal unramified extension of the power series field K = IFp((t)) is given by T = iFp((t)), where iFp is the algebraic closure of IFp, and the maximal tamely ramified extension by T({~ I mEN, (m,p) = I}).

Exercise 4. Let v be a nonarchimedean valuation of the field K and let v be an extension to the separable closure j{ of K. Then the decomposition field Zjj of v over K is isomorphic to the henselization of K with respect to v, in the sense of §6, p.I43.

§ 10. Higher Ramification Groups

The inertia group and the ramification group inside the Galois group of valued fields are only the first terms in a whole series of subgroups that we are now going to study. We assume that L I K is a finite Galois extension and that v K is a discrete normalized valuation of K, with positive residue field characteristic p, which admits a unique extension w to L. We denote by VL = ew the associated normalized valuation of L.

(10.1) Definition. For every real number s ~ -1 we define the s-th ramification group of L I K by

G s = G s (L I K) = { a E G I vdaa - a) ~ s + 1 for all a EO} .

§ 10. Higher Ramification Groups 177

Clearly, G -1 = G, Go is the inertia group I = I (L IK), and G1 the ramification group R = R(LIK) which have already been defined in (9.3). As

v£{r- 10"ra - a) = VL( r- 1(O"ra - ra») = VL (O"(ra) - ra)

and rO = 0, the ramification groups form a chain

G=G-1 2Go 2G1 2G2 2···

of normal subgroups of G. The quotients of this chain satisfy the

(10.2) Proposition. Let JrL EO be a prime element of L. For every integer s ~ 0, the mapping

JrL G /G ~ U(s)/U(s+l)

s s+l L L '

is an injective homomorphism which is independent of the prime element JrL.

Here uis) denotes the s-th group of principal units of L, i.e., uiO) = 0*

and uis ) = 1 + Jrf 0, for s ~ 1.

We leave the elementary proof to the reader. Observe that one has U (O)/U(l) '" 1 * d U(s)/U(S+l) '" l' 1 Th f G /G L L = /\. an L L = 'A, lor s ~. e actors s s+l are therefore abelian groups of exponent p, for s ~ 1, and of order prime to p, for s = O. In particular, we find again that the ramification group R = G1 is the unique p-Sylow subgroup in the inertia group I = Go.

We now study the behaviour of the higher ramification groups under change of fields. If only the base field K is changed, then we get directly from the definition of the ramification groups the following generalization of (9.5).

(10.3) Proposition. If K' is an intermediate field of L I K, then one has, for all s ~ - 1, that

Gs(LIK') = Gs(LIK) n G(LIK').

Matters become much more complicated when we pass from L I K to a Galois subextension L' I K. It is true that the ramification groups of L I K are mapped under G(LIK) -+ G(L'IK) into the ramification groups of L'IK, but the indexing changes. For the precise description of the situation we need some preparation. We will assume for the sequel that the residue field extension 'AIK of LIK is separable.


(10.4) Lemma. The ring extension 0 of 0 is monogenous, i.e., there exists an x E 0 such that 0 = O[X].

Proof: As the residue field extension )..IK is separable by assumption, it admits a primitive element X. Let I(X) E o[X] be a lifting of the minimal polynomiall(X) of X. Then there is a representative x E 0 of x such that :rr = I(x) is a prime element of O. Indeed, if x is an arbitrary representative, then we certainly have v£(/(x» 2: 1 because l(x) = O. If x itself is not as required, i.e., if v£(/(x» 2: 2, the representative x +:rrL will do. In fact, from Taylor's formula

I(x +:rrd = I(x) + 1'(X):rrL + b:rrl, bE 0,

we obtain v£(/(x + :rr£» = 1 since /'(x) E 0*, because l'(x) "10. In the proof of (6.8), we saw that the

xj:rr i = x j I (x)i , j = 0, ... , I - 1, i = 0, ... , e - 1,

form an integral basis of 0 over o. Hence indeed 0 = o[x]. o

For every a E G we now put

iLIK(a) = v£(ax - x),

where 0 = o[x]. This definition does not depend on the choice of the generator x and we may write

Gs(LIK) = {a E G I iLIK(a) 2: s + 1} . Passing to a Galois subextension L'IK of LIK, the numbers hIK(a) obey the following rule.

(10.5) Proposition. If e' = e L IL' is the ramii1cation index of L I L', then

1 iL'IK(a') = -, L iLIK(a).

e ITIL,=ql

Proof: For a' = 1 both sides are infinite. Let a' "I 1, and let 0 = o[x] and 0' = o[y], with 0' the valuation ring of L'. By definition, we have

e'iL'IK(a') = v£(a'y - y), iLIK(a) = v£(ax - x).

We choose a fixed a E G = G(LIK) such that aiL' = a'. The other elements of G with image a' in G' = G(L'IK) are then given by a., • E H = G (L I L'). It therefore suffices to show that the elements

a = ay - y and b = n (au - x) r:eH

generate the same ideal in O.


Let I(X) E a'eX] be the minimal polynomial of x over L'. Then I(X) = O'rEH(X - rx). Letting a act on the coefficients of I, we get the polynomial (af)(X) = O'rEH(X - arx). The coefficients of al - 1 are all divisible by a = ay - y. Hence a divides (af)(x) - I(x) = ±b.

To show that conversely b is a divisor of a, we write y as a polynomial in x with coefficients in 0, y = g(x). As x is a zero of the polynomial g(X) - y E a'eX], we have

g(X) - y = I(X)h(X) , h(X) E a/[X].

Letting a operate on the coefficients of both sides and then substituting X = x yields y - ay = (af)(x)(ah)(x) = ±b(ah)(x), i.e., b divides a. 0

We now want to show that the ramification group Gs(LIK) is mapped onto the ramification group G t (L'I K) by the projection

G(LIK) -+ G(L'IK),

where t is given by the function 'YJLIK : [-1,00) --* [-1, (0),

r dx t = 'YJLIK(S) = 10 (Go: Gx )

Here (Go : Gx ) is meant to denote the inverse (Gx : Go)-I when -1 ::s x ::s 0, i.e., simply 1, if -1 < x ::s O. For 0 < m ::s S ::s m + 1, mEN, we have explicitly

1 'YJLIK(S) = - (gl + g2 + ... + gm + (s - m)gm+d, gi = #Gi.

go

The function 'YJLIK can be expressed in terms of the numbers iLIK(a) as follows:

(10.6) Proposition. 'YJLIK (s) = io LUEG mint iLIK (a), S + I} - 1.

Proof: Let O(s) be the function on the right-hand side. It is continuous and piecewise linear. One has 0(0) = 'YJLIK(O) = 0, and if m 2: -1 is an integer and m < S < m + 1, then

I 1 {I } 1 I o (s) = -# a E G iLIK(a) 2: m +2 = (G . G ) = 'YJLIK(S), go o· m+J

Hence 0 = 'YJLIK. o


(10.7) Theorem (HERBRAND). Let L'I K be a Galois subextension of L I K and H = G(L IL'). Then one has

Gs(LIK)H/H = Gt(L'IK) where t = rJLIU(s).

Proof: Let G = G(L IK), G' = G(L'IK). For every a' E G', we choose an preimage a E G of maximal value iLl K (a) and show that

Let m = iLIK(a). If r E H belongs to Hm- 1 = Gm-1(LIL'), then iLIK (r) ?: m, and iLIK (ar) ?: m, so that iLIK (ar) = m. If r rf. Hm- 1,

then iLIK(r) < m and iLIK(ar) = iLIK(r). In both cases we therefore find that iLIK(ar) = min{iLIK(r), m}. Applying (10.5), this gives

1 iUIK(a') = --; L min{ iLIK(r),m}.

e r:EH

Since iLIK(r) = iLlu(r) and e' = eLIU = #Ho, (10.6) gives the formula (*), which in tum yields

a' E GsH / H {=:} iLIKCa) - 1 ?: S {=:} rJLIUCiLIK (a) - 1) ?: rJLIU(s)

{=:} iUIK(a') - 1 ?: rJLIU(s)

{=:} a' E Gt(L'IK), t = rJLIU(s). o

The function rJLIK is by definition strictly increasing. Let the inverse function be 1/tLIK : [-1, (0) -+ [-1, (0). One defines the upper numbering of the ramification groups by

Gt(LIK):= Gs(LIK) where S = 1/tLIK(t).

The functions rJLIK and 1/tLIK satisfy the following transitivity condition:

(10.8) Proposition. If L' I K is a Galois subextension of L I K, then

rJLIK = rJUIK 0 rJLIL' and 1/tLIK = o/L!L' 0 1/tL'IK.

Proof: For the ramification indices of the extensions L I K, L' I K, L I L' we have eLIK = eL'IKeLIL'· From (10.7), we obtain GslHs = (G/H)t, t = rJLIL'(s); thus

1 1 1 --#Gs = --#(G/H)t--#Hs. eLIK eL'IK eL!L'


This equation is equivalent to

1]~IK(S) = 1]~'IK(t)1]~IL'(S) = (1]L'IK 0 1]L!L')'(S).

As 1]LIK(O) = (1]L'IK 0 1]LIL')(O), it follows that 1]LIK = 1]L'IK 01]LIL' and the forn1Ula for 0/ follows. 0

The advantage of the upper numbering of the ramification groups is that it is invariant when passing from L 1 K to a Galois subextension.

(10.9) Proposition. Let L'I K be a Galois subextension of L 1 K and H = G(LIL'). Then one has

Gt(LIK)HjH = Gt(L'IK).

Proof: We put s = o/L'IK(t), G' = G(L'IK), apply (10.7) and (10.8), and get

Gt H j H = G1/ILIK(t)H j H = G~LIL'(1/ILIK(t)) = G''''LIL,(1/ILIL1(S))

= G's = Gil. o

Exercise 1. Let K = Qp and Kn = K (0, where ~ is a primitive pn -th root of unity. Show that the ramification groups of K n I K are given as follows:

Gs=G(KnIK) fors=O,

Gs = G(KnIKI) for 1::s s::s p -1,

Gs = G(KnIK2) for p::s s::s p2 -1,

Gs = 1 for pn-I ::s s.

Exercise 2. Let K' be an intermediate field of L I K. Describe the relation between the ramification groups of L I K and of L I K I in the upper numbering.

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