[Grundlehren der mathematischen Wissenschaften] Algebraic Number Theory Volume 322 || Abstract Class...

Chapter IV

Abstract Class Field Theory

§ 1. Infinite Galois Theory

Every field k is equipped with a distinguished Galois extension: the separable closure klk. Its Galois group Gk = G(klk) is called the absolute Galois group of k. As a rule, this extension will have infinite degree. It does, however, have the advantage of collecting all finite Galois extensions of k. This is why it is reasonable to try to give it a prominent place in Galois theory. But such an attempt faces the difficulty that the main theorem of Galois theory does not remain true for infinite extensions. Let us explain this in the following

Example: The absolute Galois group GF p = G (IF p IlF p) of the field IF p with p elements contains the Frobenius automorphism <p which is given by

x<fJ = x P for all x E lFp .

The subgroup (<p) = {<pn I n E Z} has the same fixed field IF p as the whole of GF p. But contrary to what we are used to in finite Galois theory, we find (<p) #- GF p. In order to check this, let us construct an element 0/ E GF p which does not belong to (<p). We choose a sequence {an}nEN of integers satisfying

an ==am modm

whenever m In, but such that there is no integer a satisfying an == a mod n for all n EN. An example of such a sequence is given by an = n' Xn, where we write n = n' pvp(n), (n', p) = 1, and 1 = n' Xn + pvp(n) Yn. Now put

If IF pm ~ IF pn, then min, so that an == am mod m, and therefore

o/n IFpm = <pan IFpm = <pam IFpm = o/m.

Observe that <p IF m has order m. Therefore the o/n define an automorphism p

0/ of lFp = U::O=llF pn. Now 0/ cannot belong to (<p) because 0/ = <pa, for a E Z, would imply 0/ IF n = <pan IF n = <pa IF n and hence an == a mod n p p p for all n, which is what we ruled out by construction.

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

262 Chapter IV. Abstract Class Field Theory

The example does not mean, however, that we have to chuck the main theorem of Galois theory altogether in the case of infinite extensions. We just have to amend it using the observation that the Galois group G = G(.Q Ik) of any Galois extension .Q 1 k carries a canonical topology. This topology is called the Krull topology and is obtained as follows. For every (1 E G we take the cosets

(1G(.QIK)

as a basis of neighbourhoods of (1, with K 1 k ranging over finite Galois subextensions of .Q Ik. The multiplication and the inverse map

G x G -+ G, «(1, r) t----+ (1r, and G -+ G, (1 t----+ (1-1,

are continuous maps, since the preimage of a fundamental open neighbourhood (1rG(.QIK), resp. (1-1G(.QIK), contains the open neighbourhood (1G(.Q IK) x rG(.Q IK), resp. (1G(.Q IK). Thus G is a topological group which satisfies the following

(1.1) Proposition. For every (finite or infinite) Galois extension .Q Ik the Galois group G = G(.Q Ik) is compact Hausdorff with respect to the Krull topology.

Proof: If (1, rEG and (1 "# r, then there exists a finite Galois subextension Klk of .Qlk such that alK "# rlK, so that aG(.QIK) "# rG(QIK) and thus (1G(.Q IK) n rG(.Q IK) = 0. This shows that G is Hausdorff. In order to prove compactness, consider the mapping

h: G -+ nG(Klk), (1 t----+ n (1IK, K K

where K Ik varies over the finite Galois subextensions. We view the finite groups G(K Ik) as discrete compact topological groups. Their product is therefore a compact topological space, by Tykhonov's theorem (see [98]). The homomorphism h is injective, because (11 K = 1 for all K is equivalent to (1 = 1. The sets U = OK#Ko G(K Ik) x {O"} form a subbasis of open sets of the product OK G (K 1 k), where Ko 1 k varies over the finite subextensions of .Q Ik and 0" E G(Ko Ik). If (1 EGis a preimage of 0", then h-1(U) = (1G(.QIKo). Thus h is continuous. Moreover h«(1G(.Q IKo» = h(G) n U, so h : G t-+ h(G) is open, and thus a homeomorphism. It therefore suffices to show that h(G) is closed in the compact set OK G(K Ik). To see this we consider, for each pair L' ;2 L of finite Galois subextensions of .Q Ik, the set

MUlL = {O (1K E 0 G(K Ik) I (1u IL = (1L} . K K

§ 1. Infinite Galois Theory 263

One clearly has h(G) = nL'::lL ML'IL. So it suffices to show that ML'IL is closed. But if G(L Ik) = {al' ... ,an}, and Si £ G(L'lk) is the set of extensions of ai to L', then

n

ML'IL = U ( Il G(K Ik) x Si x {ad) , i=l K -j-L, L'

i.e., ML'IL is indeed closed. o

The main theorem of Galois theory for infinite extensions can now be formulated as follows.

(1.2) Theorem. Let Q Ik be a (finite or infinite) Galois extension. Then the assignment

K 1---+ G(QIK)

is a I-I-correspondence between the subextensions K I k of Q I k and the closed subgroups of G (Q I k). The open subgroups of G (Q I k) correspond precisely to the finite subextensions of Q Ik.

Proof: Every open subgroup of G(Q Ik) is also closed, because it is the complement of the union of its open cosets. If K Ik is a finite subextension, then G(QIK) is open, because each a E G(QIK) admits the open neighbourhood aG(QIN) £ G(QIK), where Nlk is the normal closure of K Ik. If K Ik is an arbitrary subextension, then

G(QIK) = nG(QIKi), i

where Ki Ik varies over the finite subextensions of K Ik. Therefore G(Q IK) is closed.

The assignment K 1-+ G(QIK) is injective, since K is the fixed field of G(QIK). To prove surjectivity, we have to show that, given an arbitrary closed subgroup H of G(Qlk), we always have

H = G(QIK),

where K is the fixed field of H. The inclusion H £ G (Q I K) is trivial. Conversely, let a E G(Q IK). If L IK is a finite Galois subextension of Q IK, then a G (Q I L) is a fundamental open neighbourhood of a in G (Q I K). The map H -+ G(L IK) is certainly surjective, because the image H has fixed field K and is therefore equal to G (L I K), by the main theorem of Galois theory for finite extensions. Thus we may choose arE H such that


r IL = aiL, i.e., r E H n aG(Q IL). This shows that a belongs to the closure of H in G(QIK), and thus to H itself, so that H = G(QIK).

If H is an open subgroup of G(Q Ik), then it is also closed, and therefore of the form H = G(QIK). But G(Qlk) is the disjoint union of the open co sets of H. Since G(Q Ik) is compact, a finite number of cosets suffices to cover the group. Thus there is only a finite number of them; H = G(Q IK) has finite index in G (Q I k), and this implies that K I k has finite degree. 0

The topological Galois groups G = G(Q Ik) have the special property that there is a fundamental system of neighbourhoods of the neutral element 1 E G which consists of normal subgroups. This property leads us to the abstract, purely group-theoretical notion of a pro finite group.

(1.3) Definition. A profinite group is a topological group G which is Hausdorff and compact, and which admits a basis of neighbourhoods of 1 E G consisting of normal subgroups.

It can be shown that the last condition is tantamount to G being totally disconnected, i.e., to the condition that each element of G is equal to its own connected component. Every closed subgroup H of G is obviously again a profinite group. The disjoint coset decomposition

G = UajH j

shows immediately that H is open if and only if the index (G : H) is finite.

Profinite groups are fairly close relatives of finite groups. They can be reconstituted rather easily from their finite quotients. For the precise description of this we need the notion of projective limit, which naturally occurs in various places in number theory and which we will introduce next.

Exercise 1. Let L Ik be a Galois extension and K Ik an arbitrary extension, both contained in a common extension Qlk. If L n K = k, then the mapping

G(LKIK) ~ G(Llk), a f-* aiL,

is a topological isomorphism, that is, an isomorphism of groups and a homeomorphism of topological spaces.

Exercise 2. Given a family of Galois extensions Kdk in .Q Ik, let K Ik be the composite of all K; I k, and K; I k the composite of the extensions K j I k such that j =f::. i. If K; n K; = k for all i, then one has a topological isomorphism

G(Klk) ~ TIG(K;lk).

§ 2. Projective and Inductive Limits 265

Exercise 3. A compact Hausdorff group is totally disconnected if and only if its neutral element admits a basis of neighbourhoods consisting only of normal subgroups.

Exercise 4. Every quotient G / H of a profinite group G by a closed normal subgroup H is a pro finite group.

Exercise 5. Let G' be the closure of the commutator subgroup of a profinite group, and Gab = G / G'. Show that every continuous homomorphism G -+ A into an abelian profinite group factorizes through Gab.

§ 2. Projective and Inductive Limits

The notions of projective, resp. inductive limit generalize the operations of intersection, resp. union. If {Xi }iEI is a family of subsets of a topological space X which for any two sets Xi, X) also contains the set Xi n X) Crespo Xi U X) ), then the projective Crespo inductive) limit of this family is simply de ined by

~ Xi = n Xi Crespo !!!¥ Xi = U Xd. iEI iEI iEI iEI

Writing i ::s j if X) S; Xi Crespo XiS; X) makes the indexing set I into a directed system, i.e., an ordered set in which, for every pair i, j, there exists a k such that i ::s k and j ::s k. In the case at hand, such a k is given by Xk = Xi n X) Crespo Xk = Xi U X). For i ::s j we denote the inclusion X) ~ Xi Crespo Xi ~ X) by lij and obtain a system {Xi, Ii)} of sets and maps. The operations of intersection and union are now generalized by replacing the inclusions Ii) with arbitrary maps.

(2.1) Definition. Let I be a directed system. A projective, resp. inductive system over I is a family {Xi, Iii Ii, j E I, i ::s j} of topological spaces Xi and continuous maps

Ii) : X) ---+ Xi, resp. Ii): Xi ---+ X),

such that one has Iii = idx; and

lik = Ii) 0 iJk, resp. lik = Ijk 0 Ii),

when i ::s j ::s k.

In order to define the projective, resp. inductive limit of a projective, resp. inductive system {Xi, Ii)}, we make use of the direct product TIiEI Xi, resp. the disjoint union Ui EI Xi.


(2.2) Definition. The projective limit

X= ~ Xi iEI

of the projective system {Xi, lij} is defined to be the subset

X = { (Xi)iEI E TI Xi I 1i}(Xj) = Xi for i.::: j} iEI

The product TIi EI Xi is equipped with the product topology. If the Xi are Hausdorff, then so is the product, and it contains in this case X as a closed subspace. Indeed, one has

X = n Xi}, i~j

where Xi} = {(XdkEI E TIk Xk I lij(Xj) = xd, so that it suffices to show the closedness of the sets X ij. Writing Pi : TIkEI X k ~ Xi for the i -th projection, the two maps g = Pi, I = Ii} 0 Pj : TIkEI Xk ~ Xi are continuous, and we may write Xij = {X E TIk Xk I g(x) = I(x)}. But in the Hausdorff case the equation g (x) = I (x) defines a closed subset. This representation X = ni~j Xi} also gives the following

(2.3) Proposition. The projective limit X = ~ Xi of nonempty compact spaces Xi is itself nonempty and compact. i

Proof: If all the Xi are compact, then so is the product TIiE! Xi, by Tykhonov's theorem, and thus also the closed subset X. Furthermore, X = ni~j Xi} cannot be the empty set if the Xi are nonempty. In fact, as the product TIi Xi is compact, there would have to be an intersection of finitely many Xi} which is empty. But this is impossible: if all indices entering into this finite intersection satisfy i, j .::: n, and if Xn E X n, then the element (Xi)iEI belongs to this intersection, where we choose Xi = lin(Xn ),

for i .::: n, and arbitrarily for all other i. 0

(2.4) Definition. The inductive limit

X = !!!¥ Xi iEI

of an inductive system {Xi, lij} is defined to be the quotient

X = (llEIXi) / '"

of the disjoint union lliEI Xi, where we consider two elements Xi E Xi and x j E X j equivalent if there exists a k ::: i, j such that

lik(Xi) = lik(Xj).


In the applications, the projective and inductive systems {Xi, Ii)} that occur will not just be systems of topological spaces and continuous maps, but the Xi will usually be topological groups, rings or modules, etc., and the Ii) will be continuous homomorphisms. In what follows, we will deal explicitly only with projective and inductive systems {Gi, gij} of topological groups. But since everything works exactly the same way for systems of rings or modules, these cases may be thought of tacitly as being treated as well.

Let {Gi, gij} be a projective, resp. inductive system of topological groups. Then the projective, resp. inductive limit

resp. G = ~ Gi iEI

is a topological group as well. The multiplication in the projective limit is induced by the componentwise multiplication in the product Di EI G i· In the case of the inductive limit, given two equivalence classes x, y E G = ~ Gi, one has to choose representatives Xk and Yk in the

iEl same G k in order to define

xy = equivalence class of XkYk.

We leave it to the reader to check that this definition is independent of the choice of representatives, and that the operation thus defined makes G into a group.

The projections Pi : DiEI Gi ~ Gi, resp. the inclusions ti Gi ~ lliEI Gi, induce a family of continuous homomorphisms

gi : G ~ Gi, resp. gi: Gi ~ G

such that gi = gil 0 gj, resp. gi = gj 0 gij, for i :::: j. This family has the following universal property.

(2.5) Proposition. If H is a topological group and

hi : H ~ Gi, resp. hi: Gi ~ H

is a family of continuous homomorphisms such that

hi = gij 0 hj, resp. hi = hj 0 gil

for i :::: j, then there exists a unique continuous homomorphism

h . H ~ G = lim G· . +-- I,

1

resp. h : G = lim Gi ~ H -:-+

1

satisfying hi = gi 0 h, resp. hi = h 0 gi, for all i E I.


The easy proof is left to the reader. A morphism between two projective, resp. inductive systems {Gi,gij} and {G~,gi) of topological groups is a family of continuous homomorphisms Ii : Gi -+ G~, i E I, such that the diagrams

Gj Ij

G', G' Ij G', ~ ~

J J J

gij 1 1 g;j, resp. gij r r g;j fi Ii

Gi ~ G~ Gi ~ G~ I I

commute for i ::: j. Such a family (li)iEI defines a mapping

f: nGi ~ nG~, resp. f: l1 Gi ~ l1 G~, iEI iEI

which induces a homomorphism

f: ~ Gi ~ ~ G~, iEI iEI

iEI iEI

resp. f: ~ Gi ~ ~ G~. iEI iEI

In this way ~,resp. ~,becomes a functor. A particularly important property of this functor is its so-called "exactness". For the inductive limit ~,exactness holds without restrictions. In other words, one has the

(2.6) Proposition. Let a : {G~, gi) -+ {G i • gij} and fJ : {Gi • gij} -+ {G'f, gij} be morphisms between inductive systems of topological groups such that the sequence

G~~GI'~G'! I I

is exact for every i E I. Then the induced sequence

is also exact.

G' ct i~ ~ GiL ~ Gi'

iEI iEI

Proof: Let G' = ~ G~, G = ~ Gi, Gil = ~ G'f. We consider the iii

commutative diagram

G~ cti G Pi G~' ~ i~ I I

1 g; 1 gi P

1 g;'

G'~ G ~ Gil.


Let x E G be such that f3(x) = 1. Then there exists an i and an Xi E Gi such that gi(Xi) = x. As

there exists j :::: i such that f3i (Xi) equals 1 in G'J. Changing notation, we may therefore assume that f3i (Xi) = 1, so that there exists Yi E G~ such that ai(Yi) = Xi. Putting Y = g;(Yi), we have a(y) = x. 0

The projective limit is not exact in complete generality, but only for compact groups, so that we have the

(2.7) Proposition. Let a : {G~,g;) -+ {Gi,gij} and f3 : {Gi,gij} -+

{G~' , g;j} be morphisms between projective systems of compact topological groups such that the sequence

G' ai G. fJi Gil i--+ 1--+ i

is exact for every i E I. Then

~ G~ ~ ~ Gi L ~ G;' iii

is again an exact sequence of compact topological groups.

Proof: Let X = (Xi)iEI E ~ Gi and f3(x) = 1, so that f3i(Xi) = 1 for i

all i E I. The preimages Yi = ai 1 (Xi) ~ G~ then form a projective system of nonempty closed, and hence compact subsets of the G;. By (2.3), this means that the projective limit Y = ~ Yi ~ ~ G~ is nonempty, and

a maps every element Y E Y to x. i i o

Now that we have at our disposal the notion of projective limit, we return to our starting point, the profinite groups. Recall that these are the topological groups which are Hausdorff, compact and totally disconnected, i.e., they admit a basis of neighbourhoods of the neutral element consisting of normal subgroups. The next proposition shows that they are precisely the projective limits of finite groups (which we view as compact topological groups with respect to the discrete topology).


(2.8) Proposition. If G is a profinite group, and if N varies over the open normal subgroups of G, then one has, algebraically as well as topologically, that

G ~ ~ GIN. N

If conversely {Gj,gij} is a projective system of finite (or even profinite) groups, then

is a profinite group.

G= ~ Gj j

Proof: Let G be a profinite group and let {Ni liE l} be the family of its open normal subgroups. We make I into a directed system by defining i :'S j if Ni 2 Nj. The groups Gi = GINi are finite since the co sets of Ni in G form a disjoint open covering of G, which must be finite because G is compact. For i :'S j we have the projections gij : G j ~ Gi and obtain a projective system {Gi, gij} of finite, and hence discrete, compact groups. We show that the homomorphism

f : G -----+ ~ Gj, CT 1---+ n CTj, CTj = CT mod Nj, jEI jEI

is an isomorphism and a homeomorphism. f is injective because its kernel is the intersection nj EI Nj, which equals {I} because G is Hausdorff and the Nj form a basis of neighbourhoods of 1. The groups

Us = n Gj x D{lG;}, j¢S jES

with S varying over the finite subsets of I, form a basis of neighbourhoods of the neutral element in njEI Gj. As f- 1(Usn ~ Gj) = njESNj, we see that f is continuous. Moreover, as G is compact, the image f(G) is closed in ~ Gj. On the other hand it is also dense. For if x = (Xj)jEI E ~ Gj, and x(Us n ~ Gj) is a fundamental neighbourhood of x, then we may choose ayE G which is mapped to Xk under the projection G ~ GIN k.

where we put N k = nj ES Nj. Then y mod Ni = Xj for all i E S, so that f(y) belongs to the neighbourhood x(Us n ~ Gj). Therefore the closed set f(G) is indeed dense in ~ Gj, and so f(G) = ~ Gj. Since G is compact, f maps closed sets into closed sets, and thus also open sets into open sets. This shows that f : G ~ ~ Gj is an isomorphism and a homeomorphism.

Conversely, let {Gj, gij} be a projective system of profinite groups. As the Gj are Hausdorff and compact, so is the projective limit G = ~ Gj,


by (2.3). If Ni varies over a basis of neighbourhoods of the neutral element in Gi which consists of normal subgroups, then the groups

US=flGixflNi, i,/S iES

with S varying over the finite subsets of I , make up a basis of neighbourhoods of the neutral element in fli EI G i consisting of normal subgroups. The normal subgroups Us n ¥!!! Gi therefore form a basis of neighbourhoods of the neutral element in ¥!!! Gi; thus ¥!!! Gi is a pro finite group. 0

Let us now illustrate the notions of profinite group and projective limit by a few concrete examples.

Example 1: The Galois group G = G(Q Ik) of a Galois extension Q Ik is a profinite group with respect to the Krull topology. This was already stated in § 1. If K Ik varies over the finite Galois subextensions of Q Ik, then, by definition of the Krull topology, G(Q IK) varies over the open normal subgroups of G. In view of the identity G(Klk) = G(Qlk)jG(QIK) and of (2.8), we therefore obtain the Galois group G(Q Ik) as the projective limit

G(Q Ik) ~ ¥!!! G(K Ik)

of the finite Galois groups G(K Ik).

Example 2: If p is a prime number, then the rings Zjpnz, n E N, form a projective system with respect to the projections Z j pn Z -+ Z j pm Z, for n ~ m. The projective limit

Zp = ¥!!! ZjpnZ n

is the ring of p-adic integers (see chap. II, § 1).

Example 3: Let 0 be the valuation ring in a p-adic number field K and pits maximal ideal. The ideals pn, n EN, make up a basis of neighbourhoods of the zero element 0 in 0.0 is Hausdorff and compact, and so is a pro finite ring. The rings 0 jpn , n EN, are finite and we have a topological isomorphism

o ~ ¥!!! ojpn, a ~ fl (a mod pn). n nEN

The group of units U = 0* is closed in 0, hence Hausdorff and compact, and the subgroups U(n) = 1 + pn form a basis of neighbourhoods of 1 E U. Thus

U ~ ¥!!! UjU(n) n

is also a pro finite group. In fact, we have seen all this already in chap. II, §4.

272 Chapter Iv. Abstract Class Field Theory

Example 4: The rings Z / nZ, n EN, form a projective system with respect to the projections Z/nZ -+ Z/mZ, nlm, where the ordering on N is now given by divisibility, nlm. The projective limit

~

Z = ¥!!! Z / nZ n

was originally called the Priifer ring, whereas nowadays it has become customary to refer to it by the somewhat curt abbreviation "zed-hat" (or "zee-hat"). This ring is going to occupy quite an important position in what follows. It contains Z as a dense subring. The groups ni, n EN, are precisely the open subgroups of i, and it is easy to verify that

ifni ~ 71,/nZ.

Taking, for each natural number n, the prime factorization n = np pVp, the Chinese remainder theorem implies the decomposition

71,/nZ ~ n 71,/ pVp71" p

and passing to the projective limit,

i ~ nZp. p

This takes the natural embedding of Z into i to the diagonal embedding Z-+ npZp, a t-+ (a,a,a, ... ).

Example 5: For the field IF q with q elements, we get isomorphisms

one for every n EN, by mapping the Frobenius automorphism qJn to 1 mod nZ. Passing to the projective limit gives an isomorphism

G(iFqllFq) ~ i

which sends the Frobenius automorphism qJ E G ( iF q IlF q) to 1 E i, and the sub~oup (qJ) = {qJn I n E Z} onto the dense (but not closed) subgroup 71,

of Z. Given this, it is now clear, in the example at the beginning of this chapter, how we were able to construct an element 1/r E G (iF q IlF q) which did not belong to (qJ). In fact, looking at it via the isomorphism G (iF q IlF q) ~ i, what we did amounted to writing down the element

( ... ,0,0, Ip, 0, 0, ... ) E n Ze = i, e

which does not belong to 71,.


Example 6: Let Q I Q be the ~ extension obtained by adjoining all roots of unity. Its Galois group G(Q IQ) is then canonically isomorphic (as a topological group) to the group of units i* ~ flp Z; of i,

G(ijIQ) ~ i*. This isomorphism is obtained by passing to the projective limit from the canonical isomorphisms

G(Q(JLn)IQ) ~ (Z/nZ)*,

where JLn denotes the group of n -th roots of unity.

~

Example 7: The groups Zp and Z are (additive) special cases of the class of procyclic groups. These are profinite groups G which are topologically generated by a single element (J; i.e., G is the closure ((J) of the subgroup ((J) = {(In I n E Z}. The open subgroups of a procyclic group G = ((J) are all of the form Gn . Indeed, Gn is closed, being the image of the continuous map G -+ G, X f-+ x n , and the quotient group G / Gn is finite, because it contains the finite group {(Jv mod Gn I 0 :::: v < n} as a dense subgroup, and is therefore equal to it. Conversely, if H is a subgroup of G of index n, then G n ~ H ~ G and n = (G : H) :::: (G : G n) :::: n, so that H = Gn.

Every procyclic group G is a quotient of the group i. In fact, if G = ((J), then we have for every n the surjective homomorphism

Z/nZ -+ G/Gn , I mod nZ f-+ (J mod G n ,

and in view of (2.7), passing to the projective limit yields a continuous surjection i -+ G.

Example 8: Let A be an abelian torsion group. Then the Pontryagin dual

X (A) = Hom(A, Q/Z)

is a profinite group. For one has

A=UAi, i

where Ai varies over the finite subgroups of A, and thus

X(A) = ~ X(Ai) i

with finite groups X(Ai). If for instance,

A =Q/Z = U ~Z/Z, nEN


then x(*Z/Z) = Z/nZ, so that ~

X(Q/Z) ~ ~ Z/nZ = Z. n

Example 9: If G is any group and N varies over all normal subgroups of finite index, then the profinite group

~

G = ~ G/N N

is called the pro/mite 5.?mpletion of G. The profinite completion of Z, for example, is the group Z = ~ Z/nZ.

n

Exercise 1. Show that, for a profinite group G, the power map G x Z -* G, (0" , n) H- O"n, extends to a continuous map

G x Z -* G, (0" , a) H- O"a ,

and that one has (O"a)b = O"ab and O"a+b = O"aO"b if G is abelian.

Exercise 2. If 0" E G and a = lim ai E Z with ai E Z, then O"a = lim O"a; is in G. i-+oo i-+oo

Exercise 3. A pro-p-group is a profinite group G whose quotients GIN, modulo all open normal subgroups N, are finite p -groups. Imitating exercise 1, mak~ sense of the powers O"a, for all 0" E G and a E Zp.

Exercise 4. A closed subgroup H of a profinite group G is called a p-Sylow subgroup of G if, for every open normal subgroup N of G, the group H N I N is a p-Sylow subgroup of GIN. Show:

(i) For every prime number p, there exists a p-Sylow subgroup of G.

(ii) Every pro-p-subgroup of G is contained in a p-Sylow subgroup.

(iii) Every two p-Sylow subgroups of G are conjugate.

Exercise S. What is the p-Sylow subgroup of Z and of Z;?

Exercise 6. If {G;} is a projective system of profinite groups and G = 1.i!!! G i , i

then Gab = 1.i!!! Gfb (see § 1, exercise 5). i

§ 3. Abstract Galois Theory 275

§ 3. Abstract Galois Theory

Class field theory is the final outcome of a long development of algebraic number theory the beginning of which was Gauss's reciprocity law

(~)(~) = (-1) a2l ¥.

The endeavours to generalize this law finally produced a theory of the abelian extensions of algebraic and p-adic number fields. These extensions L I K are classified by certain subgroups NL = N L I K A L of a group A K attached to the base field. In the local case, AK is the multiplicative group K* and in the global case it is a modification of the ideal class group. At the heart of this theory there is a mysterious canonical isomorphism

which - if we view things in the right way - encapsulates the reciprocity law in its most general form. Now, this map can be abstracted completely from the field-theoretic situation and treated on a purely group theoretical basis. In this way, class field theory can be given an abstract, but elementary foundation, to which we will now turn.

We begin our considerations by giving ourselves a profinite group G. The theory we are about to develop is purely group theoretical in nature. However, the only applications we have in mind are field theoretical, and the language of field theory allows immediate insights into the group theoretical relations. We will therefore formally interpret the profinite group G as a Galois group in the following way. (Let us remark in passing that every profinite group is indeed the Galois group G = G(klk) of a Galois field extension k Ik; this will allow the reader to rely on his standard knowledge of Galois theory whenever the formal development in terms of group theory alone would seem odd.)

We denote the closed subgroups of G by G K, and call these indices K "fields"; K will be called the fixed field of G K. The field k such that G k = G is called the base field, and k denotes the field satisfying Gl( = {l}. The

field belonging to the closure «(J') of the cyclic group «(J') = {(J'k IkE Z) generated by an element (J' EGis simply called the fixed field of (J'.

We write formally K S; L or L I K if G L S; G K, and we call the pair L I K a field extension. L I K is called a finite extension, if G L is open, i.e., of finite index in G K, and this index

[L : K] := (GK : Gd


will be called the degree of L I K. L I K is said to be normal or Galois if G L

is a normal subgroup of G K. If this is the case, we define the Galois group of LIK by

G(LIK) = GK/GL.

If N ;2 L ;2 K are Galois extensions of K, we define the restriction of an element a E G(NIK) to L by

aiL = a mod G(NIL) E G(LIK).

This gives a homomorphism

G(NIK) ----+ G(LIK), a 1---+ aiL,

with kernel G (N I L). The extension L I K is called cyclic, abelian, solvable, etc., if the Galois group G(L IK) has these properties. We put

("intersection")

if G K is topologically generated by the subgroups G Ki ' and

K = fl Ki ("composite") i

if GK = ni GKi • If GK' = a-1GKa for a E G, we write K' = K U •

Now let A be a continuous multiplicative G-module. By this we mean a multiplicative abelian group A on which the elements a E G operate as automorphisms on the right, a : A ---+ A, a f-+ aU. This action must satisfy

(i) a1 =a,

(ii) (ab)U = aU bU,

(iii) aUT = (aU) T ,

(iv) A = U[K:kl<OO AK,

where A K in the last condition denotes the fixed module A G K under G K ,

so that AK = {a E A I aU = a for all a E GK} ,

and where K varies over all extensions that are finite over k. The condition (iv) says that G operates continuously on A, i.e., the map

G x A ----+ A, (a,a) 1---+ aU,

is continuous, where A is equipped with the discrete topology. Indeed, this continuity is equivalent to the fact that, for every element (a, a) E G x A, there exists an open subgroup U = G K of G such that the neighbourhood a U x {a} of (a, a) is mapped to the open set {au}, and this means simply that aU E AU = AK.


Remark: In the exponential notation aa , the operation of G on A appears as an action on the right. This notation is adequate for many computations in the case of multiplicative G-modules A. For instance, the notation aa-I := aa a-I is to be preferred to writing (a - l)a = O'a . a-I. On the other hand, classical usage often calls for an operation on the left. Thus in the case of a Galois extension L I K of actual fields, the Galois group G (L I K) acts as the automorphism group on L from the left, and therefore also in the same way on the multiplicative group L *. This occasional switch from the left to the right should not confuse the reader.

For every extension LIK we have AK ~ AL, and if LIK is finite, then we have the norm map

NLIK : AL --+ AK, NLIK(a) = rIaa , a

where a varies over a system of representatives of G L \ G K • If L I K is Galois, then AL is a G(LIK)-module and one has

A G(LIK) - A L - K·

At the center of class field theory there is the norm residue group

HO( G(LIK), Ad = AK INLIK AL.

We also consider the group

H- I (G(L IK), Ad = NLIK ALI IG(LIK)AL,

where NLIKAL = {a E AL I NLIK(a) = I}

is the "norm-one group" and IG(LIK)AL is the subgroup of NL1KAL which is generated by all elements

with a E AL, and a E G(LIK). If G(LIK) is cyclic and a is a generator, then IG(LIK)AL is simply the group

A~-I = {aa-I I a E Ad .

In fact, the formal identity O' k - 1 = (l + 0'+ ... + O'k-I)(O' - 1) implies ak_1 ba - I . h b rIk- 1 a i

a = WIt = i =0 a .

Let us now apply the notions introduced so far to the example of Kummer theory. For this, we impose on the G-module A the following axiomatic condition.


(3.1) Axiom. One has H-1 (G(L IK), Ad = 1 for all finite cyclic extensions L I K .

The theory we are about to develop makes reference to a surjective G -homomorphism

g;y : A --+ A, a 1---+ a P ,

with finite cyclic kernel /Lp. The order n = #/Lp is called the exponent of the operator g;y. The case of prime interest to us is when g;y is the n-th power map a f-+ an, and /Lp = /Ln = {~ E A I ~n = I} is the group of "n-th roots of unity" in A.

We now fix a field K such that /Lp £;; AK. For every subset B £;; A, let K (B) denote the fixed field of the closed subgroup

H = {a E G K I ba = b for all b E B}

of G K. If B is G K -invariant, then K (B) IK is obviously Galois. A Kummer extension (with respect to g;y) is by definition an extension of the form

where Ll £;; A K. A Kummer extension K (g;y -I (Ll» I K is always Galois, and its Galois group is abelian of exponent n. Indeed, for an extension K (g;y-I (a))lK, we have the injective homomorphism

G(K(g;y-I(a»IK) ~ /Lp, a 1---+ aa-I,

where a E g;y-I (a). Since /Lp £;; AK, this definition does not depend on the choice of a. Thus, for a Kummer extension L = K(g;y-I(Ll» =

TIaELl K (g;y-I (a», the composite map

G(LIK) --+ TI G(K(g;y-I(a»IK) --+ /L~ aELl

is an injective homomorphism.

The following proposition says that conversely, any abelian extension L I K of exponent n is a Kummer extension.

(3.2) Proposition. If L IK is an abelian extension of exponent n, then

L=K(g;y-l(Ll») with Ll=A~nAK.

If in particular, L I K is cyclic, then we find L = K (a) with a P = a E A K .


Proof: We have ,p-l(L1) S;;; AL, for if x E A and x~ = a~ = a E AK, a E AL, then x = ~a E AL for some ~ E f.L~ S;;; AK. Therefore K(,p-l(L1)) S;;; L. On the other hand, the extension LIK is the composite of its cyclic subextensions. For it is the composite of its finite subextensions, and the Galois group of a finite subextension is the product of cyclic groups, which may be interpreted as Galois groups of cyclic subextensions. Let now M I K be a cyclic subextension of L I K. It suffices to show that M S;;; K(,p-l(L1)). Let U be a generator of G(MIK) and ~ a generator of f.L~. Let d = [M : K], d' = njd and ~ = ~dl. Since NMIK(~) = ~d = 1, (3.1) shows that ~ = a CT - 1 for some a E AM. Thus K S;;; K(a) S;;; M. But aCT; = ~ia. Thus aCT; = a is equivalent to i == 0 mod d, so that K (a) = M. But (a~)CT-l = (aCT-l)~ = ~~ = 1, so that a = a~ E AK; then a E ,p-l(L1), and therefore M S;;; K(,p-l(L1)). 0

As the main result of general Kummer theory, we now obtain the following

(3.3) Theorem. The correspondence

..1 ~ L = K(,p-l(L1))

is a I-I-correspondence between the groups ..1 such that A ~ S;;; ..1 S;;; A K and the abelian extensions L IK of exponent n.

If ..1 and L correspond to each other, then Ar n AK = ..1, and we have a canonical isomorphism

L1jA~ ~Hom(G(LIK),IL~), amodA~~Xa,

where the character Xa : G(L IK) --+ IL~- is given by Xa(u) = aCT-I, for a E ,p-l(a).

Proof: Let LIK be an abelian extension of exponent n. By (3.2), we then find L = K (,p -1 (..1)) with ..1 = A rnA K. We consider the homomorphism

..1-+ Hom(G(LIK),IL~), a ~ Xa,

where Xa(u) = aCT-I, a E ,p-l(a). Since

Xa = 1 {:=:::} a CT - 1 = 1 for all U E G(L IK)

{:=:::} a E AK {:=:::} a = a~ E A~,

it has the kernel A~. To prove the surjectivity, we let X E Hom(G(L IK), f.L~). X defines a cyclic extension M I K and is the composite of homomorphisms

G(LIK) --+ G(MIK) L IL~. Let U be a generator of G(MIK). Since


NMIKCx(a» = x(a)[M:K] = 1, we deduce from (3.1) that x(a) = au-I for some a E AM. Now, (as:»U-1 = (aU-I)s:> = x(a)S:> = 1, so that a = as:> E

A~ n AK = ..1. For r E G(LIK), one has x(r) = x(r 1M) = al:- I = Xa(r), so that X = Xa. This proves the surjectivity, and we obtain an isomorphism

..1/A~ ~ Hom(G(LIK),JLs:».

If ..1 is any group between A~ and AK and if L = K(p-I(..1)), then ..1 = A~ n AK. In fact, putting ..1' = A~ n AK, we have just seen that one has

..1'/A~ ~ Hom(G(L IK), JLs:» .

The subgroup ..1 / A ~ corresponds under Pontryagin duality to the subgroup Hom(G(LIK)/H,JLs:», where

H = { a E G (L I K) I Xa (a) = 1 for all a E ..1} .

As au-I = Xa(a) for a E p-I(a), H leaves fixed the elements of p-I(..1), and as K(p-I(..1)) = L, we find that H = 1, so that Hom(G(LIK)/H,JLs:» = Hom(G(LIK),JLs:». It follows that ..1/A~ = ..1' / A~, i.e., ..1 = ..1'.

It is therefore clear that the correspondence ..1 t-+ L = K (p -I (..1)) is a I-I-correspondence, as claimed. This finishes the proof of the theorem. 0

Remarks and Examples: 1) If LIK is infinite, then Hom(G(LIK),JLs:» has to be interpreted as the group of all continuous homomorphisms X : G(LIK) --+ JLs:>, i.e., as the character group of the topological group G(LIK).

2) The composite of two abelian extensions of K of exponent n is again of the same type, and all of them lie in the maximal abelian extension of exponent n. It is given by K = K (p -I (AK », and for the Pontryagin dual

G(KIK)* = Hom ( G(KIK), Q/Z) = Hom ( G(KIK), JLs:»

we have by (3.3) that

3) If k is an actual field of positive characteristic p and k is the separable closure of k, then A may be chosen to be the additive group k and p to be the operator

p : k -+ k, a t---+ pa = aP - a.

Then axiom (3.1) is indeed satisfied, for we have, in complete generality:


(3.4) Proposition. For every cyclic finite field extension L I K, one has H-1 (G(L IK), L) = 1.

Proof: The extension L IK always admits a normal basis {O"c 10" E G(L IK)}, so that L = EBu K O"C. This means that L is a G(L IK)-induced module in the sense of §7, and then H-1(G(LIK),L) = 1, by (7.4). 0

The Kummer theory with respect to the operator tJa = a P - a is usually called Artin-Schreier theory.

4) The chief application of the theory developed above is to the case where G is the absolute Galois group G(klk) of an actual field k, A is the multiplicative group k* of the algebraic closure, and tJ is the n-th power map a 1--+ an, for some natural number n which is relatively prime to the characteristic of k (in particular, n is arbitrary if char(k) = 0). Axiom (3.1) is always satisfied in this case and is called Hilbert 90 because this statement occurs as Satz number 90 among the 169 theorems in Hilbert's famous "Zahlbericht" [72]. Thus we have the

(3.5) Theorem (Hilbert 90). For a cyclic field extension L I K one always has

In other words: An element ot E L* of norm NLIK(ot) = 1 is of the form ot = f3u- 1,

where f3 E L * and 0" is a generator of G(L IK).

Proof: Let n = [L : K]. By virtue of the linear independence of the automorphisms 1,0", ... , O"n-l (see [15], chap. 5, § 7, no. 5), there exists an element Y E L * such that

1+ 2 1+ + + n-2 n-l f3 = Y + otyu + ot u yU + ... + ot u ... u yU #- o.

As NLIK(ot) = 1, one gets otf3u = f3, and thus ot = f31-U. o

If now the field K contains the group ILn of n-th roots of unity, the operator tJ(a) = an has exponent n, and we obtain the following corollary, which is the most important special case of theorem (3.3).


(3.6) Corollary. Let n be a natural number which is relatively prime to the characteristic of the field K, and assume that J-Ln ~ K.

Then the abelian extensions L I K of exponent n correspond 1-1 to the subgroups ..1 ~ K * which contain K M, via the rule

and we have

Hilbert's theorem 90, which is the main basis of this corollary, admits the following generalization to arbitrary Galois extensions L I K , which goes back to the mathematician EMMY NoEl' HER (1882-1935). Let G be a finite group and A a multiplicative G -module. A l-cocycle, or crossed homomorphism, of G with values in A is a function f : G --+ A satisfying

f(ar:) = f(a)r: f(r:)

for all a, r: E G. The l-cocycles form an abelian group Z 1 (G , A). For every a E A, the function

is a l-cocycle, for one has

fa(ar:) = aur:-I = (au-I)r:ar:-I = fa (a)r: fa(r:).

The functions fa are called l-coboundaries and form a subgroup B 1 (G , A) of ZI(G, A). We define

HI(G, A) = ZI(G, A)/B1(G, A)

and obtain as a first result about this group the

(3.7) Proposition. If G is cyclic, then HI (G , A) ;:; H -I (G , A).

Proof: Let G = (a). If f E ZI(G, A), then for k ~ 1

k-I . f(a k) = f(ak-I)U f(a) = f(a k- 2)U2 f(a)U f(a) = ... = n f(a)(11 ,

and f(l) = 1 because f(l) = f(1)f(1). If n = #G, then

n-I .

Nof(a) = n f(a)(11 = f(a n ) = f(1) = 1, i=O

i=O


so that f(a) E NGA = {a E A I NGa = rr:d aai = I}. Conversely we obtain, for every a E A such that NGa = I, a I-cocycle by putting I(a) = a and

k-I . I(a k ) = n aa l

•

i=O

The reader is invited to check this. The map 1 t-+ 1 (a) therefore is an isomorphism between Z 1 (G , A) and N GA. This isomorphism maps B 1 (G , A)

onto I G A, because fEB 1 (G, A) {=:::::} 1 (a k ) = aak-I for some fixed a {=:::::}

I(a) = aa-I {=:::::} f(a) E IGA. 0

Noether's generalization of Hilbert's theorem 90 now reads:

(3.8) Proposition. For a finite Galois field extension L I K, one has that

HI(G(LIK),L*) = 1.

Proof: Let f : G -+ L * be a I-cocycle. For eEL *, we put

a = L I(a)ca . aEG(LIK)

Since the automorphisms a are linearly independent (see [15], chap. 5, § 7, no. 5), we can choose eEL * such that a =j:. O. For r E G(L IK), we obtain

a' = Lf(a)'ca, = LI(r)-I/(ar)ca, = I(r)-Ia , a a

. I() R,-I . h R -I I.e., r = p Wit p = a . o

This proposition will only be applied once in this book (see chap. VI, (2.5)).

Exercise 1. Show that Hilbert 90 in Noether's formulation also holds for the additive group L of a Galois extension L I K . "

Hint: Use the normal basis theorem.

Exercise 2. Let k be a field of characteristic p arid k its separable closure. For fixed n:::: 1, consider in the ring of Witt vectors W(k) (see chap. II, §4, exercise 2--6) the additive group Wn(k) of truncated Witt vectors a = (aO,al, ... ,an-I). Show that axiom (3.1) holds for the G(klk)-module A = Wn(k).

Exercise 3. Show that the operator

gJ : Wn(k) --+ Wn(k) , gJa = Fa - a,

is a homomorphism with cyclic kernel f.t1fJ of order pn. Discuss the corresponding Kummer theory for the abelian extensions of exponent pn.


Exercise 4. Let G be a profinite group and A a continuous G -module. Put

HI(G, A) = ZI(G, A)/B1(G, A),

where ZI(G, A) consists of all continuous maps 1 : G ~ A (with respect to the discrete topology on A) such that l(a7:) = I(a)~ 1(7:), and BI(G, A) consists of all functions of the form la(a) = au-I, a E A. Show that if g is a closed normal subgroup of G, then one has an exact sequence

I ~ H1(G/g,Ag) ~ H1(G,A) ~ H1(g,A).

Exercise 5. Show that HI(G, A) = ~ HI(G/N, AN), where N varies over all the open normal subgroups of G.

Exercise 6. If 1 ~ A ~ B ~ C ~ 1 is an exact sequence of continuous G-modules, then one has an exact sequence

1 ~ AG ~ BG ~ CG ~ H1(G,A) ~ H1(G,B) ~ H1(G,C).

Remark: The group H1(G,A) is only the first term of a whole series of groups Hi(G, A), i = 1,2,3, ... , which are the objects of group cohomology (see [145]). Class field theory can also be built upon this theory (see [10], [108]).

Exercise 7. Even for infinite Galois extensions LIK, one has Hilbert's theorem 90: H1(G(LIK),L*) = 1.

Exercise 8. If n is not divisible by the characteristic of the field K and if I1-n denotes the group of n-th roots of unity in the separable closure K, then

HI(GK ,l1-n) ~ K*/KM.

§ 4. Abstract Valuation Theory

The further development will now be based on a fixed choice of a surjective continuous homomorphism

d:G-+Z

from the profinite group G onto the procyc1ic group Z = ~ ZjnZ (see §2, example 4). This homomorphism will produce a theory which is an abstract reflection of the ramification theory of p-adic number fields. Indeed, in the case where G is the absolute Galois group Gk = G(k/k) of a p-adic number field k, such a surjective homomorphism d : G ~ Z arises via the maximal unramified extension k / k: if IF q is the residue class field of k, then, by chap. II, § 9, p. 173 and example 5 in § 2, we have canonical isomorphisms

G(k/k) ~ G(iFq /lFq) ~ Z

§ 4. Abstract Valuation Theory 285

which associate to the element 1 E Z the Frobenius automorphism qJ E G(klk). It is defined by

a<fJ == aq mod p for a E 0,

where 0, resp. p, denote the valuation ring of k, resp. its maximal ideal. The homomorphism d : G ~ Z in question is then given, in this concrete case, as the composite

G ---* G(klk) ~ Z.

In the abstract situation, the initial choice of a surjective homomorphism d : G ~ Z mimics the p -adic case, but the applications of the theory are by no means confined to p-adic number fields. The kernel I of d has a certain fixed field klk, and d induces an isomorphism G(klk) ~ Z.

More generally, for any field K we denote by h the kernel of the restriction d : G K ~ Z, and call it the inertia group over K. Since

h = G K n I = G K n G'k = G Kk'

the fixed field K of h is the composite

K=Kk.

We call K I K the maximal unramified extension of K. We put

fK=(Z:d(GK)), eK=(I:h)

and obtain, when fK is finite, a surjective homomorphism

1 .-.. dK = -d : GK ---* Z

fK

with kernel h, and an isomorphism

dK : G(KIK) ~ Z.

(4.1) Definition. The element qJ KEG (K I K) such that dK (qJ K) = 1 is called the Frobenius over K.

For a field extension L IK we define the inertia degree /LIK and the ramification index eLIK by

/LIK = (d(GK): d(GL)) and eLIK = (h : h).

For a tower of fields K ~ L ~ M this definition obviously implies that

fMIK = /LIK fMIL and eMIK = eLIK eMIL·


(4.2) Proposition. For every extension L I K we have the "fundamental identity"

[L : K] = /LIK eLIK .

Proof: The exact commutative diagram

1 1 1

immediately yields, if L IK is Galois, the exact sequence

1 ~ hilL ~ G(LIK) ~ d(GK)ld(Gd ~ 1.

If LIK is not Galois, we pass to a Galois extension MIK containing L, and get the result from the above transitivity rules for e and f. 0

LIK is called unramified if eLIK = 1, i.e., if L S; K. LIK is called totally ramified if /LIK = 1, i.e., if L n K = K. In the unramified case, we have the surjective homomorphism

G(KIK) ~ G(LIK)

and, if fK < 00, we call the image ({JLIK of ({JK the Frobenius automorphism ofLIK.

For an arbitrary extension L I K one has ~ ~

L=LK,

since LK = LKk = Lk = L, and L n KIK is the maximal unramified subextension of L I K. It clearly has degree

fLIK = [L nK: K].

Equally obvious is the

(4.3) Proposition. If fK and /L are finite, then /LIK = /LlfK, and we have the commutative diagram

In particular, one has ({JL Ii = ({Ji:L1K •

§ 4. Abstract Valuation Theory 287

The Frobenius automorphism governs the entire class field theory like a king. It is therefore most remarkable that in the case of a finite Galois extension L IK, every a E G(L IK) becomes a Frobenius automorphism once it is manreuvered into the right position. This is achieved in the following manner. For what follows, let us assume systematically~ that fK < 00.

We pass from the Galo~ extension L I K to the extension L I K and consider in the Galois group G (L I K) the semigroup

Frob(LIK) = {a E G(LIK) I dKCa) EN} .

Observe here that dK : GK ----+ Z factorizes through G(LIK) because G I = I L S; I K ; recall also that 0 1. N. Firstly, we have the

(4.4) Proposition. For a finite Galois extension L IK the mapping

Frob(L IK) ----+ G(L IK), a ~ aiL,

is surjective.

Proof: Let a E G(L IK) and let <p E G(L IK) be an element such that dK(<p) = 1. Then <PIK = <PK and <pILniL = <PLnKIK. Restricting a to the

maximal unramified subextension L n K I K, it becomes a power of the ~roben~s automorphism, al LnK = <P2nKIK' so we may choose n in N. As

L = LK, we have ~ ~ ~

G(LIK) ~ G(LIL n K).

If now r E G(L IK) is mapped to a<p-n IL under this isomorphism, then a = r<pn is an element satisfying aiL = r<pn IL = a<p-n<pn IL = a and al K = <P'k. Hence dK(a) = n, and so a E Frob(LIK). 0

Thus e~ery element a E G(L IK) may be lifted to an element a E Frob(L I K). The following proposition shows that this lifting, considered over its fixed field, is actually the Frobenius automorphism.

(4.5) Proposition. Let a E Frob(LIK), and let E be the fixed field of a. Then we have:

(i) f2:IK = dK(a), (ii) [E: K] < 00, (iii) E = L, (iv) a = <P2:.


Proof: (i) JJ n K is the fixed field of aiR = cp~K(a-), so that

/EIK = [JJ n K : K] = dda).

~ ,-..,J,......",......

(ii) One has K ~ JJ K = E ~ L; thus ,......" ,...... ,...... ,......"

eEIK = (h : IE) = #G(JJIK) ::: #G(LIK)

is finite. Therefore [JJ : K] = /EIKeEIK is finite as well. .......

(iii) The canonical surjection r = G(L I JJ) -+ G(JJ I JJ) = Z has to be bijective. For since r = (a) is procyc1ic, one finds (r : rn) ::: n for every n E N (see § 2, p. 273). Thus the induced maps r j rn ~ i j ni are bijective and so is r -+ Z. But GellJJ) = G(EIJJ) implies that l = E. (iv) /EIKdE(a) = dK(a) = /EIK; thus dE(a) = 1, and so a = CPE. 0

Let us illustrate the situation described in the last proposition by a diagram, which one should keep in mind for the sequel.

L 'PL

L=JJ

I

~ JJ o-='PE

hlK K K

fElK 'PK

All the preceding discussions arose entirely from the initial datum of the homomorphism d : G -+ i. We now add to the data a multiplicative Gmodule A, which we equip with a homomorphism that is to play the role of a henselian valuation.

(4.6) Definition. A henselian valuation of Ak with respect to d : G -+ i is a homomorphism

satisfying the following properties:

(i) V(Ak) = Z;2 Z and ZjnZ ~ ZjnZ for all n EN,

(ii) v(N KlkAK) = /K Z for all finite extensions K Ik.

Exactly like the original homomorphism d : G k -+ i, the henselian valuation v : Ak -+ i has the property of reproducing itself over every finite extension K of k.

§ 4. Abstract Valuation Theory

(4.7) Proposition. For every field K which is finite over k, the fonnula

defines a surjective homomorphism satisfying the following properties:

(i) VK = VK" oa for all a E G.

(ii) For every finite extension L IK, one has the commutative diagram

VL ..-AL )z

NLIK 1 1 iLlK

VK ..-AK ) z.

289

Proof: (i) If r runs through a system of representatives of G k / G K, then a-I r a sweeps across a system of representatives of G k / a-I G K a = Gk/GK". Hence we have, for a E AK,

(ii) For a E AL one has:

(4.8) Definition. A prime element of AK is an element JrK E AK such that VK(JrK) = 1. We put

For an unramified extension L IK, that is, an extension such that /LIK = [L : K], we have from (4.7), (ii) that VL IAK = VK. In particular, a prime element of AK is itself also a prime element of AL. If on the other hand, L IK is totally ramified, i.e., JLIK = 1, and if JrL is a prime element of AL, then JrK = NLIK(Jr[) is a prime element of AK.


Exercise 1. Assume that every closed abelian subgroup of G is procyclic. Let K Ik be a finite extension. A microprime p of K is by definition a conjugacy class (cr) <; G K of some Frobenius element cr E Frob(kIK) which is not a proper power crm, n > I, of some other Frobenius element cr' E Frob(kIK). Let spec(K) be the set of all microprimes of K. Show that if L IK is a finite extension, then there is a canonical mapping

rr : spec(L) ~ spec(K).

Above any microprime p there are only finitely many microprimes SlJ of L, i.e., the set rr-1(p) is finite. We write SlJlp to mean SlJ E rr-1(p).

Exercise 2. For a finite extension L I K and a microprime SlJI p of L, let f'rJIP = d(SlJ)/d(p). Show that

I: f'rJIP = [L : K]. 'rJlp

Exercise 3. For an infinite extension L I K, let

spec(L) = ~ spec(La), a

where La I K varies over the finite subextensions of L I K. What are the microprimes of k?

Exercise 4. Show that if LIK is Galois, then the Galois group G(LIK) operates transitively on spec(L). The "decomposition group"

G'rJ(LIK) = {cr E G(LIK)Is,p" = SlJ} is cyclic, and if Z'rJ = LG~(LIK) is the "decomposition field" of SlJ E spec(L), then LIZ'rJ is unramified.

§ 5. The Reciprocity Map

Continuing with the notation of the previous section, we consider again a pro finite group G, a continuous G -module A, and a pair of homomorphisms

d:G~Z, V:Ak~Z,

such that d is continuous and surjective and v is a henselian valuation with respect to d. In the following we introduce the convention that the letter K, whenever it occurs without embellishments or commentary to the contrary, will always denote a field of finite degree over k. We furthermore impose the following axiomatic condition, which will be systematically assumed in the sequel.

(5.1) Axiom. For every unramified finite extension L IK one has

Hi (G(LIK), Ud = 1 for i = 0, - 1.

§ 5. The Reciprocity Map

For an infinite extension L I K we set

NLIKAL = nNMIKAM, M

with M I K varying over the finite subextensions of L I K .

Our goal is to define a canonical homomorphism

rLIK : G(LIK) ~ AKINLIKAL

291

for every ~ite Galois extension L I K. To this end, we pass from L I K to the extension L I K and define first a mapping on the semigroup

Frob(LIK) = {a E G(LIK) I dK(a) E H} .

(5.2) Definition. The reciprocity map

rLIK : Frob(LIK) ~ AKINLIKAL

is defined by rLIK(a) = NEIK(JrE) modNLIKAL,

where E is the fixed field of a and JrE E AE is a prime element.

Observe that E is of finite degree over K by (4.5), and a becomes the Frobenius automorphism ({JE over E. The definition of rLIK(a) does not depend on the choice of the element JrE. For another one differs from JrE

only by an element U E UE, and for this we have NEIK(U) E NLIKAL' so that N E I d u) E N M I K A M for every finite Galois subextension M I K of LIK. To see this, we may clearly assume that E ~ M. Applying (5.1) to the unramified extension MIL', one finds U = NMIE(t:), t: E UM, and thus

NEIK(U) = NEIK(NMIE(t:») = NMIK(t:) E NMIKAM.

Next we want to show that the rec~rocity map rLIK is multiplicative. To do

this, we consider for every a E G(L IK) and every n E H the endomorphisms

a ~ aCT - 1 = aCT la, n-l .

I

a ~ aCTn = n aCT . i=O

In formal notation, this gives an = :n ~ 11, and we find that

(a - 1) 0 an = an 0 (a - 1) = an - 1.

Now we introduce the homomorphism

N = NLIK : AL ~ AK

and prove two lemmas for it.


(5.3) Lemma. Let q;, a E Frob(L IK) with ddq;) = 1, dda) = n. If E is the fixed field of a and a EAr, then

N rlK (a) = (N 0 CPn)(a) = (CPn 0 N)(a).

Proof: The maximal unramified subextension EO = En K I K is of degree n, and its Galois group G(EoIK) is generated by the Frobenius automorphism CProlK = CPKlro = CPIKlro = cplro. Consequently, NrO IK = CPnIAL'O' On the other hand, one has E K = L and E n K = EO, and therefore N rlro = N IAL" For a E Ar we thus get

NrIK(a) = NroIK(Nrlro(a») = N(a)f{Jn = N(af{Jn).

~ I"-.J ~ ~

The last equation follows from cpG(LIK) = G(LIK)cp. o

The subgroup IO(iIK)Ui' which is generated by all elements of the

form U,-I, U E Ui' r E G(LIK), is mapped to 1 by the homomorphism N = NilK : Ui --+ UK' We therefore obtain an induced homomorphism

N : Ho(G(LIK), Ui) --+ UK

on the quotient group Ho(G(L IK), Ui) = Uri IO(iIK)Ui' For this group, we have the following lemma.

~ ~ ~

(5.4) Lemma. If x E Ho(G(L IK), Ui) is fixed by an element cP E G(L IK) such that dK(cp) = 1, i.e., xf{J = x, then

Proof: Let x = U mod IO(iIK)Ui' with xf{J-l = 1, so that

r f{J-l TI 'i-1

U = u j , 7:j E G(L IK). j=1

Let M I K be a finite Galois subextension of L I K. In order to prove that N(u) E NMIKUM, we may assume that U,Uj E UM and L S; M. Let n = [M : K], a = q;n and let E :2 M be the fixed field of a. Further, let En I E be the unramified extension of degree n, i.e., the fixed field of an = cp1;. By (5.1), we can then find elements U, Uj E Urn such that

U = Nrnlr(u) = u an , Ui = Nrnlr(Ui) = urn.

§5. The Reciprocity Map 293

By (*), the elements urp-I and nj u7; -I only differ by an element x E U En such that N En IE (X) = 1. Hence - again by (5.1) - they differ by an element of the form y<1-I, with y E U En' We may thus write

-rp-I - _rpn_1 n -,;-1 - (-rpn)rp-I n -,;-1 U - Y U j - Y U j •

j j

Applying N gives N(u)rp-I = N(yrpn)rp-I , so that

N(u) = N(yrpn) . z,

for some z E UK such that zrp-I = 1; therefore zrp = z, and z E UK. Finally, applying O'n and putting Y = y<1n = NEnIE(Y) E UE, we obtain, observing n = [M : K] and using (5.3), that

N(u) = N(u)<1n = N(yrpn)<1nz<1n = N(yrpn)zn

= NEIK (y)NMIK (z) E NMIKUM. o

(5.5) Proposition. The reciprocity map

ri lK : Frob(LIK) ---+ AK/NIIKAI

is multiplicative.

Proof: Let 0'10'2 = 0'3 be an equation in Frob(LIK), nj = dK(O'j), Ej the fixed field of O'j and rrj E AE; a prime element, for i = 1,2,3. We have to show that

NEIiK (rrl)NE2IK (rr2) == NE3IK(rr3) mod NIIKAI'

Choose a fixed cP E G(LIK) such that dK(CP) = 1 and put

rj = O'j-Icpn; E G(LIK).

From 0'10'2 = 0'3 and nl + n2 = n3, we then deduce that

r3 = 0';10'1-lcpn2+nl = 0'2-lcpn2(cp-n20'Icpn2)-lcpnl.

n2 n2 Putting 0'4 = cp-n20'Icpn2, n4 = dK(0'4) = nl, E4 = Ef ,rr4 = rrf E AE4 and r4 = 0'4- lcpn4, we find r3 = r2r4 and

NE4IK(rr4) = NEIIK(rrl).

We may therefore pass to the congruence


the proof of which uses the identity T3

NEilK(:rrj) = N(JTtn;). Thus, if we put T2T4. From (5.3), we have

then the congruence amounts simply to the relation N(u) E NLIKAL. For this, however, lemma (5.4) gives us all that we need.

n' I u.-1({Jn; -I I Since CPn; 0 (cp - 1) = cpn; - 1 and JTt 1- = JTj 1 = JTj';- ,we have

From the identity T3 = T2T4, it follows that (T3 - 1) + (1 - T2) + (1 - T4) = (1 - T2)(1 - T4). Putting now

we obtain 4

.0-1 T1 ,·-1 u'" = u/ . j=2

For the element x = u mod IG(LIK)UL E Ho(G(LIK), UL), this means that

X({J-I = 1, and so x({J = x; then by (5.4), we do get N(u) = N(x) E NLIKAL.

o

From the surjectivity of the mapping

Frob(LIK) --+ G(LIK)

(5.6) Proposition. For every finite Galois extension L I K, there is a canonical homomorphism

given by

rLIK((}) = NEIK(JTE) mod NLIK AL,

where E is the fixed field of a preimage jj E Frob(L IK) of () E G(L IK) and JT E E A E is a prime element. It is called the reciprocity homomorphism ofLIK.

§5. The Reciprocity Map 295

Proof: We first show that the definition of rLIK(a) is independent of the choice of the preimage a E Frob(LIK) of a. For this, let a' E Frob(LIK) be another preimage, E' its fixed field and JrIJI E AIJI a prime element. If dK (a) = dK (a'), then a Ii = a' Ii and aiL = a'IL, so that a = a', and there is nothing to show. However, if we have, say, dK(a) < dK(a'), then a' = ai for some i E Frob(LIK), and ilL = 1. The fixed field E"

of i contains L, so rIIK(i) == NIJ"IK(JrIJ") == 1 mod NLIKAL. It follows therefore that rIIK(a') = rIIK(a)rIIK(i) = rIIK(a).

The fact that the mappin$ is a homomorphism now follows directly from (5.5): if ai, a2 E Frob(L I K) are preimages of ai, a2 E G (L I K), then a3 = al a2 is a preimage of a3 = al a2. 0

The definition of the reciprocity map expresses the fundamental principle of class field theory to the effect that Froben~s automorphisms correspond to prime elements: the element a = ({JIJ E G~IE) is map'ped to JrIJ E AIJ; for reasons of functoriality, the inclusion G(L I E) "--+ G(L IK) 'corresponds to the norm map N IJIK : AIJ ---+ AK. So the definition of rLIK(a) is already forced upon us by these requirements. This principle appears at its purest in the

(5.7) Proposition. If LIK is an unramified extension, then the reciprocity map

is given by rLIK«({JLIK) = JrK mod NLIKAL,

and is an isomorphism.

Proof: In this case one has L = K and ({JK E G(KIK) is a preimage of ({JLIK with fixed field K, i.e., rLIK«({JLIK) = JrK mod NLIKAL. The fact that we have an isomorphism is seen from the composite

G(LIK) -+ AK/NLIKAL -+ Z/nZ ~ 7l/n71,

with n = [L : K], where the second map is induced by the valuation VK : AK ---+ Z because vK(NLIKAd ~ nZ. It is an isomorphism, for if vK(a) == 0 mod nZ, then a = uJr~n, and since u = NLIK(S) for some S E UL, by (5.1), we find a = NLIK(SJr~) == 1 mod NLIK AL. On the side of the homomorphisms, the generators ({JLIK, JrK mod NLIK AL, and 1 mod nZ correspond to each other, and everything is proved. 0

The reciprocity homomorphism rLIK exhibits the following functorial behaviour.


(5.8) Proposition. Let LIK and L'IK' be finite Galois extensions, so that K ~ K' and L ~ L', and let a E G. Then we have the commutative diagrams

G(L'IK') rL'lK'

) AK'lNulK,Au G(LIK) rLIK

) AKINLIKAL

1 1 NK'IK 1 a- la

G(LIK) rLIK

) AKINLIKAL G(LaIKa) r£<IIKa

AKa IN £<r IKa Av" ~

where the vertical arrows on the left are given by a' f--+ a' I L, resp. by the conjugation r f--+ a-I r a .

Proof: Let a' E G(L'IK') and a = a'IL E G(LIK). If fi' E Frob(L'IK') is a preimage of a', then fi = fi' Ii E Frob(LI K) is a preimage of a such that dK(fi) = fK'IKdK,(fi') E N. Let E' be the fixed field of fi'. Then IJ = E' n L = IJ' n 17 is the fixed field of fi and fLJ'ILJ = 1. If now np E ALJ' is a prime element of E', then nLJ = NpILJ(nLJ) E ALJ is a prime element of IJ. The commutativity of the diagram on the left therefore follows from the equality of norms

NLJIK(nLJ) = NLJIK(NLJ'ILJ(nLJ')) = NLJ'ldnLJ') = NK'IK(NLJ'IK,(nLJ')).

On the other hand, let r E G(L IK), and let f be a preimage in Frob(LIK) with fixed field E, and f EGa lifting of f to k. Then E a is the fixed field of a-I fa lia, and if n E ALJ is a prime element of E, then n a E ALJa is a prime element of IJa. The commutativity of the diagram on the right therefore follows from the equality of norms

o

Another very interesting functorial property of the reciprocity map is obtained via the transfer (Verlagerung in German). For an arbitrary group G, let G' denote the commutator subgroup and write

Gab = GIG'

for the maximal abelian quotient group. If then H ~ G is a subgroup of finite index, we have a canonical homomorphism

Ver : Gab --+ Hab,

which is called transfer from G to H. This homomorphism is defined as follows (see [75J, chap. IV, § 1).

§ 5. The Reciprocity Map 297

Let R be a system of representatives for the left cosets of H in G, G = R H, 1 E R. If a E G we write, for every pER,

and we define

a p = p' a p , apE H , p' E R ,

Ver(a mod G') = TI ap mod H'. pER

Another description of the transfer results from the double coset decomposition

of G in terms of the subgroups (a) and H. Letting f (r) denote the smallest natural number such that aT = r-laf(T)r E H, one has Hn(r-lar) = (aT), and we find that

Ver(a mod G') = TI aT mod H'. T

This formula is obtained from the one above by choosing for R the set {air Ii = 1, ... , fer)}. Applying this to the reciprocity homomorphism

rLIK : G(LIK)ab ----* AK/NLIKAL

we get the

(5.9) Proposition. Let L IK be a finite Galois extension and K' an intermediate field. Then we have the commutative diagram

G(L IK,)ab rL1K '

AK,/NLIK,AL ---+

Ver r r G(L IK)ab

rLiK AK/NLIKAL, ---+

where the arrow on the right is induced by inclusion.

Proof: Let us write temporarily G = G(LIK) ~d H G(LIK'). Let a E G(LIK)~and let a be a preimage in Frob(LIK) with fixed field E and S = G (L I E) = (a). We consider the double coset decomposition G = ljSrH and put ST = r-lSr n H and aT = Claf(T)r as above. Let

T

G=G(LIK), H=G(LIK'), S=(a), f=rlL and aT=aTIL. Then we obviously have


and therefore

Ver( a mod G(L IK)') = TI a, mod G(L IK')'. , For every r, let w, vary over a system of right coset representatives of H / ST' Then one has

H = US,w, and G = U SrwT •

Let ET be the fixed field of if" i.e., the fixed field of ST' lJ' is the fixed fiel~ of r-lifr so that lJ, I E' is the unramified subextension of degree !(r) in LIE'. If now rr E A.E is a prime element of lJ, then rr' E A Er is a prime element of gr, and thus also of E T • In view of the above double coset decomposition, we therefore find

NEIK(rr) = TI rr'Wr = TI(TI(rrT)Wr) = TINE'rIK,(rr') , T,Wr r Wr r

and since if, E Frob(LIK') is a pre image of a, E G(LIK'), it follows that

rLIK(a) == TIrLIKI(a,) == rLIKI(TI a,) == rLIKI(Ver(a mod G(LIK)')). T ,

o

Exercise 1. Let L I K be abelian and totally ramified, and let 1r be a prime element of AL. If then a E G(LIK) and

y\?-I = JTa-l

with y E Ul, then rLlK(a) = N(y) mod NL1KAL, where N = NllK (B. D WORK , see [122], chap. XIII, § 5).

Exercise 2. Generalize the theory developed so far in the following way. Let P be a set of prime numbers and let G be a pro-P -group, i.e., a profinite group all of whose quotients GIN by open normal subgroups N have order divisible only by primes in P.

Let d : G --+ Zp be a surjective homomorphism onto the group Zp = DpEP Zp, and let A be a G-module. A henselian P-valuation with respect to d is by definition a homomorphism

v: Ak --+ Zp

which satisfies the following properties:

(i) v(AK ) = Z ;2 Z and ZlnZ ;:::: ZlnZ for all natural numbers n which are divisible only by primes in P,

(ii) v(NK1kAK) = fKZ for all finite extensions Klk, where!K = (d(G) : d(GK)).

Under the hypothesis that Hi (G(L IK), U d = 1 for i = 0, -1, for all unramified extensions L IK, prove the existence of a canonical reciprocity homomorphism rLIK : G(LIK)ab --+ AKINL1KAL for every finite Galois extension LIK.

§ 6. The General Reciprocity Law 299

Exercise 3. Let d : G ---+ Z be a surjective homomorphism, A a G-module which satisfies axiom (5.1), and let v : Ak ---+ Z be a henselian valuation with respect to d.

Let K Ik be a finite extension and let spec(K) be the set of microprimes of K (see §4, exercise 1-5). Define a canonical mapping

rK : spec(K) ---+ Ad NklK Ak ,

and show that, for a finite extension, the diagram

spec(L) rL

AL/NkILAk --+

rrl 1 NLIK

spec(K) rK

AK/NkIKA"k --+

commutes. Show furthermore that, for every finite Galois extension L I K, rK induces the reciprocity isomorphism

rLIK : G(LIK) ---+ AK/NLIKAL.

Hint: Let cP E G K be an element such that dK (cp) EN. Let E be the fixed field of cP and

where K", I K varies over the finite subextensions of ElK, and where the projective limit is taken with respect to the norm maps N Kfi IKa : AKfi ---+ AKa. Then there is a

surjective homomorphism Vl: : Al: ---+ Z and a homomorphism Nl:IK : Al: ---+ AK.

§ 6. The General Reciprocity Law

We now impose on the continuous G -module A the following condition.

(6.1) Class Field Axiom. For every cyclic extension L IK one has

for i = 0,

for i = -1.

Among the cyclic extensions there are in particular the unramified ones. For them the above condition amounts precisely to requiring axiom (5.1), so that one has

(6.2) Proposition. For a finite unramified extension L IK, one has

Hi(G(LIK),UL) =1 for i=O,-l.


Proof: Since LIK is unramified, a prime element lrK of AK is also a prime element of AL. As H-I(G(LIK), Ad = 1, every element U E UL such that NLIKCu) = 1 is of the form u = aa-I, with a E AL, (f = C{JLIK. So writing a = Slr'K, S E U L, we obtain u = sa-I. This shows that H-I(G(LIK), Ud = 1.

On the other hand, the homomorphism VK : AK ~ Z gives rise to a homomorphism

VK : AK /NLIK AL ---+ Z/nZ = 7L/n7L,

where n = [L : K] = iLIK, because vK(NLIKAd = iLlKZ = nZ. This homomorphism is surjective as VK(lrK mod NLIK Ad = 1 mod nZ, and it is bijective as #AK / N LIK AL = n. If now u E UK, then we have u = N LIK (a), with a E AL, since vKCu) = O. But 0 = VK(U) = vKCNLIK(a» = nvLCa), so we get in fact a E UL. This proves that HO(G(L IK), Ud = 1. 0

By definition, a class field theory is a pair of homomorphisms

(d: G ~ i, v: A ~ i), where A is a G-module which satisfies axiom (6.1), d is a surjective continuous homomorphism, and v is a henselian valuation. From proposition (6.2) and §5, we obtain for every finite Galois extension L IK, the reciprocity homomorphism

rLIK : G(LIK)ab ---+ AK/NLIKAL.

But the class field axiom yields moreover the following theorem, which represents the main theorem of class field theory, and which we will call the general reciprocity law.

(6.3) Theorem. For every finite Galois extension L IK, the homomorphism

rLIK : G(LIK)ab ---+ AK/NLIKAL

is an isomorphism.

Proof: If MIK is a Galois subextension of LIK, we get from (5.8) the commutative exact diagram

1---+ G(LIM) -----+) G(LIK) -----+) G(MIK) ~ 1

1 rLIM

We use this diagram to make three reductions.


First reduction. We may assume that G(L IK) is abelian. For if the theorem is proved in this case, then, putting M = Lab the maximal abelian subextension of L IK, we find G(L IK)ab = G(M IK), and the commutator subgroup G(L 1M) of G(L IK) is precisely the kernel of rLIK, i.e., G(LIK)ab -+ AK/NLIKAL is injective. The surjectivity follows by induction on the degree. Indeed, in the case where G(L IK) is solvable, one has either M = L or [L : M) < [L : KJ, and if rMIK and rLIM are surjective, then so is rLIK. In the general case, let M be the fixed field of a p-Sylow subgroup. MIK need not be Galois, but we may use the left part of the diagram, where rLIM is surjective. It then suffices to show that the image of NMIK is the p-Sylow subgroup Sp of AK/NLIKAL. That this holds true for all p amounts to saying that rLIK is surjective. Now the inclusion AK ~ AM induces a homomorphism

i : AK/NLIKAL ~ AM/NLIMAL,

such that NMIK 0 i = [M : K). As ([M : KJ, p) = 1, Sp [M:K\ Sp is

surjective, so Sp lies in the image of N MIK, and therefore of rLIK.

Second reduction. We may assume that L I K is cyclic. For if M I K varies over all cyclic subextensions of L I K , then the diagram shows that the kernel of rLIK lies in the kernel of the map G(LIK) -+ TIM G(MIK). Since G(L IK) is abelian, this map is injective and hence the same is true of rLIK. Choosing a proper cyclic subextension M I K of L I K , surjectivity follows by induction on the degree as in the first reduction for solvable extensions.

Third reduction. Let LLK be cyclic. We may assume that !LIK = 1. To see this, let M = L n K be the maximal unramified subextension of L IK. Then !LiM = 1 and rMIK is an isomorphism by (5.7). In the bottom sequence of our diagram, the map NMIK is injective because the groups in this sequence have the respective orders [L : M), [L : KJ, [M : K) by axiom (6.1). Therefore rLIK is an isomorphism if rLIM is.

Now let LIK be cyclic and totally ramified, i.e., !LIK = 1. Let a _be_a generator of G(L IK). We view a via the isomorphism G(L IK) ~ G(L IK) as an element of G(LIK\ and obtain the element a = acpL E Frob(LIK), which is a preimage of a E G(L IK) such that dK (a) = dK(cpr) = !LJ5 = 1. We thus find for the fixed field ElK of a that f.J; I K = 1, and so E n K = K. Let M I K be a finite Galois subextension of L I K containing E and L, let MO = M n K be the maximal unramified subextension of M I K, and put N = N MIMo. As JEIK = !LIK = 1, one finds N IAE = N ElK, N IAL = NLIK (see the proof of (5.3)).


For the injectivity of rLIK, we have to prove this: if rLIK (o-k) = 1, where o .:::::: k < n = [L : K), then k = O.

In order to do this, let nI; E AI;, nL E AL be prime elements. Since E,L ~ M, nI; and nL are both prime elements of M. Putting n1 = unf, u E U M, we obtain

rLIK(o-k) == N(n1) == N(u)· N(nf) == N(u) mod NLIK AL.

From rLIK(o-k) = 1, it thus follows that N(u) = N(v) for some v E UL, so that N(u-1v) = 1. From axiom (6.1), we may write u-1v = aO"-1 for some a E AM, and find in AM the equation

(nfv)a-l = (nfvi- l = (n1u-1v)a-1 = (aO"-I)'T-I = (aa-I)O"-I,

and so x = nfva l - a E AMo. Now vMO(x) E i and nVMo(x) = VM(X) = k imply that one has k = 0, and so rLIK is injective. The surjectivity then follows from (6.1). 0

The inverse of the mapping rLIK : G(LIK)ab --+ AK/NLIKAL gives, for every finite Galois extension L I K, a surjective homomorphism

( ,LIK): AK --* G(LIK)ab

with kernel N L IK AL. This map is called the norm residue symbol of L I K. From (5.8) and (5.9) we have the

(6.4) Proposition. Let LIK and L'IK' be finite Galois extensions such that K ~ K' and L ~ L', and let 0- E G. Then we have the commutative diagrams

AK' ,L'IK'» G(L'IK,)ab AK ( ,LIK)

G(LIK)ab

NK'IK 1 1 0"1 1 0"* AK

,LIK) G(LIK)ab, AKa

,LalKa) ) G(L a IKO")ab, )

and if K' ~ L, we have the commutative diagram

AK' ,LIK')

G(L IK,)ab )

r rver AK

,LIK) G(LIK)ab. )


The definition of the nonn residue symbol automatically extends to infinite Galois extensions L 1 K. For if L ilK varies over the finite Galois subextensions, then

G(LIK)ab = ~ G(LiIK)ab i

(see §2, exercise 6). As (a,Li'IK)ILab = (a,LiIK) for Li' ;2 Li, the I

individual nonn residue symbols (a, Li IK), a E AK, detennine an element

(a,LIK) E G(LIK)ab.

In the special case of the extension K 1 K, we~ find the following intimate connection between the maps dK, VK, and ( ,K IK).

(6.5) Proposition. One has

(a, K IK) = fP~K(a) , and thus dK 0 ( ,K IK) = VK.

Proof: Let L 1 K be the subextension of K 1 K of degree j. As Z / j Z = Z/jZ, we have vda) = n + jz, with nEZ, z E Z; that is, a = uJr'Kbf, with U E UK, bEAK. From (5.7), we obtain

(a, K IK) IL = (a, L IK) = (JrK, L IK)n(b, L IK)f = fPLIK = fP~K(a) IL .

~ v (a) Thus we must have (a, K IK) = fPKK . D

The main goal of field theory is to classify all algebraic extensions of a given field K. The law governing the constitution of extensions of K is hidden in the inner structure of the base field K itself, and should therefore be expressed in tenns of entities directly associated with it. Class field theory solves this problem as far as the abelian extensions of K are concerned. It establishes a I-I-correspondence between these extensions and certain subgroups of A K. More precisely, this is done as follows.

For every field K, we equip the group AK with a topology by declaring the co sets aN L IK A L to be a basis of neighbourhoods of a E A K , where L 1 K varies over all finite Galois extensions of K. We call this topology the norm topology of AK.

(6.6) Proposition. (i) The open subgroups of AK are precisely the closed subgroups of finite index.

(ii) The valuation VK : AK ~ Z is continuous.


(iii) If L I K is a finite extension, then N L IK : AL ---* AK is continuous.

(iv) AK is Hausdorff if and only if the group

A~ = nNLIKAL L

of universal norms is trivial.

Proof: (i) If N is a subgroup of AK, then

N=AK" U aN. aN-;N

Now, if N is open, so are all co sets aN, so that N is closed, and since N has to contain one ofthe neighbourhoods NLIK AL of the basis of neighbourhoods of 1, N is also of finite index. If, conversely, N is closed and of finite index, then the union of the finitely many co sets aN #- N is closed, and so N is open. (ii) The groups ii, fEN, form a basis of neighbourhoods of 0 E i (see § 2), and if L I K is the unramified extension of degree f, then it follows from (4.7) that

which shows the continuity of v K .

(iii) Let NMIK AM be an open neighbourhood of lEAK. Then

NLIK(NMLILAMd = NMLIKAML ~ NMIKAM,

which shows the continuity of NLIK.

(iv) is self-evident.

The finite abelian extensions L I K are now classified as follows.

(6.7) Theorem. Associating

L t----+ NL = NLIK AL

D

sets up a 1-I-correspondence between the finite abelian extensions L I K and the open subgroups N of AK. Furthermore, one has

L1 ~ L2 {:=::} NLJ ;2 NL2' NLJL2 = NLJ n N L2 , NLJnL2 = NLJNL2'

The field L corresponding to the subgroup N of AK is called the class field associated with N. By (6.3), it satisfies

G(LIK) ~ AKIN.


Proof of (6.7): If L 1 and L2 are abelian extensions of K, then the transitivity of the norm implies NL, L2 ~ NL, nNL2' If, conversely, a E NL, nNL2 , then the element (a, LIL2IK) E G(LIL2IK) projects trivially onto G(Li IK), that is, (a, Li IK) = 1 for i = 1,2. Thus (a, LIL2IK) = 1, i.e., a E N L,L2' We therefore have NL,L2 = N L, n NL2' and so

N L, 2 NL2 {:=} N L, nNL2 = NL,Lz = NLz {:=} [LIL2 : K]

= [L2 : K] {:=} LI ~ L 2.

This shows the injectivity of the correspondence L 1--+ NL .

If N is any open subgroup, then it contains the norm group NL = N L IK AL of some Galois extension LIK. (6.3) implies that NL = NUb, so we may assume LIK to be abelian. But (N,LIK) = G(LIL') for some intermediate field L' of L I K. Since N 2 N L , the group N is the full preimage of G(L IL') under the map ( ,L IK) : AK ~ G(L IK), and thus it is the full kernel of ( ,L'IK) : AK ~ G(L'IK). Thus N = N u . This shows that the correspondence L 1--+ NL is surjective.

Finally, the equality NL,nLz = NL,NLz is obtained as follows. LI n L2 ~ Li implies that NL,nLz 2 N Li , i = 1,2, and thus

NL,nLz 2 NL,NLz .

As NL,NLz is open, we have just shown that N L,NL2 = NL for some finite abelian extension L I K. But NLi ~ NL implies L ~ Lin L 2, so that

o

Exercise 1. Let n be a natural number, and assume the group /.Ln = {~ E A I ~n = I} is cyclic of order n, and A S; An. Let K be a field such that /.Ln S; AK, and let L I K be the maximal abelian extension of exponent n. If L I K is finite, then one has NL1KAL = AK· Exercise 2. Under the hypotheses of exercise 1, Kummer theory and class field theory yield, via Pontryagin duality G(L IK) x Hom(G(L IK), /.Ln) -+ /.Ln, a nondegenerate bilinear mapping (the abstract "Hilbert symbol")

( , ): AKIAK x AKIAK -7/.Ln·

Exercise 3. Let p be a prime number and (d : G -+ 7lp , v : Ak -+ 7l p ) a p-class field theory in the sense of § 5, exercise 2. Let d' : G -+ tl p be another surjective homomorphism, and K'IK the 7lp -extension defined by d'. Let v' : Ak -+ tl p be the composite of

Ak ( ,K'IK» G(K'IK) ~ 7l p •

Then (d', v') is also a p-class field theory. The norm residue symbols associated to (d, v) and (d', v') coincide. (No analogous statement holds in the case of i-class field theories (d : G -+ i, v : Ak -+ i).)


Exercise 4. (Generalization to infinite extensions.) Let (d : G """* Z, v : Ak """* Z) be a class field theory. We assume that the kernel Uk of Vk : Ak """* Z is compact for every finite extension K Ik. For an infinite extension K Ik, put

AK = ~ AKa'

where Ka Ik varies over the finite subextensions of K Ik and the projective limit is taken with respect to the norm maps NKplKa : AKp """* AKa' Show:

1) For every (finite or infinite) extension L I K, one has a norm map

NL1K : AL -+ AK ,

and if L I K is finite, there is an injection irlK : h """* AL • If furthermore L I K is Gal ' th h A~ ~ A-;-G(LIK)

OlS, en one as K = L •

2) For every extension K Ik with finite inertia degree h = [K nf : k], (d, v) induces a class field theory (dK : GK -+ Z, VK : AK """* Z). 3) If K ~ K' are extensions of k with h, h, < 00, and LIK and L'IK' are (finite or infinite) Galois extensions with L ~ L', then one has a commutative diagram

h, (,L'IK'» G(L'IK,)ab

NK'IK 1 1 AK (,LIK» G(LIK)ab.

Exercise 5. If LIK is a finite Galois extension, then G1b is a G(LIK)-module in a canonical way, and the transfer from G K to G L is a homomorphism

Ver: G't """* (G~b)G(LIK) .

Exercise 6. (Tautological class field theory.) Assume that the profinite group G satisfies the condition: for every finite Galois extension,

Ver: G,;! """* (G1b)G(LIK)

is an isomorphism. (These are the profinite groups of "strict cohomological dimension 2" (see [145], chap. III, tho (3.6.4».) Put AK = G't and form the direct limit A = ~ AK via the transfer. Then AK is identified with AGK.

Show that for every cyclic extension L I K one has

#Hi(G(LIK) A ) = {[L : K] for ~ = 0, ,L 1 for I = -1,

and that for every surjective homomorphism d : G """* Z, the induced map v : Ak = Gab """* Z is a henselian valuation with respect to d. The corresponding reciprocity map rLIK : G(LIK) """* AK/NL1KAL is essentially the identity.

Abstract class field theory acquires a much broader range of applications if it is generalized as follows.

Exercise 7. Let G be a profinite group and B(G) the category of finite G-sets, i.e., of finite sets X with a continuous G -operation. Show that the connected, i.e., transitive G-sets in B(G) are, up to isomorphism, the sets G/GK, where GK is an open subgroup of G, and G operates via multiplication on the left.


If X is a finite G -set and x EX, then

Jl'1(X,X) = Gx = {a E G lax = x}

is called the fundamental group of X with base point x. For a map I : X ~ Y in B(G), we put

G(XIy) = Auty(X).

I is called Galois if X and Y are connected and G (X I Y) operates transitively on the fibres I-I(y).

Exercise 8. Let I : X ~ Y be a map of connected finite G-sets, and let x EX, Y = I (x) E Y. Show that I is Galois if and only if Jl'1 (X, x) is a normal subgroup of Jl'1 (Y, y). In this case, one has a canonical isomorphism

G(XIY) ~ Jl'1(Y,y)/Jl'I(X,x).

A pair of functors A = (A*,A*): B(G) ~ (ab) ,

consisting of a contravariant functor A* and a covariant functor A* from B(G) to the category (ab) of abelian groups is called a double functor if

A*(X) = A*(X) =: A(X)

for all X E B(G). We define

AK = A(G/GK).

If I: X ~ Y is a morphism in B(G), then we put

A*(f) = f* and A*(f) = 1*·

A homomorphism h : A ~ B of double functors is a family of homomorphisms heX) : A(X) ~ B(X) representing natural transformations A* ~ B* and A* ~ B*.

A G -modulation is defined to be a double functor A such that (i) A(X l:J Y) = A(X) x A(Y). (ii) If among the two diagrams

X g' X' g'* +---- A(X) -------+ A(X')

11 11' and 1*1 1 I; Y

g Y' g* +---- A(Y) -------+ A(yl)

in B(G), resp. (ab), the one on the left is cartesian, then the one on the right is commutative.

Remark: G-modulations were introduced in a general context by A. DRESS under the name of Mackey functors (see [32]).

Exercise 9. G -modulations form an abelian category.

Exercise 10. If A is a G-module, then the function A(G/GK ) = AGK extends to a G -modulation A in such a way that, for an extension L I K, the map f* : AK ~ AL ,

resp. 1* : AL ~ AK , induced by I : G / G L ~ G / G K, is the inclusion, resp. the norm NL1K •


The rule A f-+ A is an equivalence between the category of G -modules and the category of G -modulations with "Galois descent", i.e., of those G -modulations A such that

f* : A(y) -+ A(X)G(X1y),

for every Galois mapping f : X -+ Y, is an isomorphism.

Exercise 11. G -modulations are explicitly given by the following data. Let Bo(G) be the category whose objects are the G-sets G j U, where U varies over the open subgroups of G, and whose morphisms are just the projections n : GjU -+ GjV for U S; V, as well as the maps c(a) : GjU -+ GjaUa- l , rU f-+ rUa- 1 = ra-l(aUa- I ), for a E G.

Let A = (A', A,) : Bo(G) -+ (ab) be a double functor and for n : GjU -+ GjV (U S; V), resp. c(a) : GjU -+ GjaUa- 1 (a E G), define

Ind~ = A,(n): A(GjU) -+ A(GjV),

Res):; = A'(n): A(GjV) -+ A(GjU),

c(a), = A,(c(a)): A(GjU) -+ A(GjaUa-I ).

If for any three open subgroups U, V S; W of G, one has the induction formula

R W I dV "I dunaVa-J () R v esu 0 n W = L... n u oc a ,0 eSvna-Jua' U\W/V

then A extends uniquely to a G-modulation A : B(G) -+ (ab).

Hint: If X is an arbitrary finite G-set, then the disjoint union

Ax = U A(GjGx ) XEX

is again a G-set, because c(a),A(GjGx ) = A(GjGax ). Define A(X) to be the group

A(X) = Homx(X,Ax)

of all G-equivariant sections X -+ Ax of the projection Ax -+ X.

Exercise 12. The function nab (G j G K) = G'fJ' extends to a G -modulation

nab: B(G) -+ (pro-ab)

into the category of pro-abelian groups. Thus, for an extension L I K, the maps f* : G'fJ' -+ G1b , resp. f, : G1b -+ G'fJ', induced by f : G j G L -+ G / G K are given by the transfer, resp. the inclusion GL -+ G K •

Exercise 13. Let A be a G -modulation. For every connected finite G -set X, let

N A(X) = n f,A(Y),

where the intersection is taken over all Galois maps f : Y -+ X. Show that the function N A(X) defines a G-submodulation N A of A, the modulation of universal norms.

Exercise 14. If A is a G -modulation, then the completion A is again a G -modulation which, for connected X, is given by

A(X) = ll!!! A(X)jf,A(y),

where the projective limit is taken over all Galois maps f : Y -+ X.


For the following, let d : G --+ Z be a fixed surjective homomorphism. Let f : X --+ Y be a map of connected finite G -sets and x EX, Y = f (x) E Y. The inertia degree, resp. the ramification index, of f is defined by

fxlY = (d(G y) : d(Gx)), resp. eXIY = (/y : Ix),

where Iy , resp. Ix, is the kernel of d : G y --+ Z, resp. d : Gx --+ Z. f is called unramified if ex IY = l.

Exercise 15. d defines a G-modulation Z such that the maps f*, f*, corresponding to a mapping f : X --+ Y of connected G-sets, are given by

-. ..-. eXIY ..-. ..-.. Z(Y) = Z ( ) Z = Z(X).

fxlY

This gives a homomorphism of G-modulations

d : nab ---+ Z. Exercise 16. An unramified map f : X --+ Y of connected finite G-sets is Galois, and d induces an isomorphism

G(XIy) ~ Z/fxlyZ.

Let f/lxIY E G(XIY) be the element which is mapped to 1 mod fxlyZ.

Let A be a G-modulation. We define a henselian valuation of A to be a homomorphism

v:A--+Z

such that the submodulation v(A) of Z comes from a subgroup Z ~ Z which contains Z and satisfies Z/nZ = Z/nZ for all n EN. Let U denote the kernel of A.

Exercise 17. Compare this definition with the definition (4.6) of a henselian valuation of a G-module A.

Exercise 18. Assume that for every unramified map f : X --+ Y of connected finite G-sets, the sequence

0--+ U(Y) ~ U(X) ~ U(X) A U(Y) --+ 0

is exact, and that A(y)[X:Y] ~ f*A(X) for every Galois mapping f : X --+ Y (the latter is a consequence of the condition which will be imposed in exercise 19). Then the pair (d, v) gives, for every Galois mapping f : X --+ Y, a canonical "reciprocity homomorphism"

rXIY : G(XIy) --+ A(Y)/f*A(X).

Exercise 19. Assume, beyond the condition required in exercise 18, that for every Galois mapping f : X --+ Y with cyclic Galois group G(XIY), one has

(A(y) : f*A(X)) = [X : Y] and ker f* = im(a* - 1),

where [X: Y] = #f-1(y), with y E Y, and a is a generator ofG(XIy). Then ifrXIY is an isomorphism for every Galois mapping f : X --+ Y of prime degree [X : y], so is

rXIY : G(Xly)ab --+ A(Y)/f*A(X),

for every Galois mapping f : X --+ Y.


Exercise 20. Under the hypotheses of exercise 18 and 19 one obtains a canonical homomorphism of G -modulations

A ~ nab

whose kernel is the G-modulation N A of universal norms (see exercise 13). It induces an isomorphism

of the completion A of A (see exercise 14).

Remark: The theory sketched above and contained in the exercises has a very interesting application to higher dimensional class field theory. In chap. V, (1.3), we will show that, for a Galois extension L IK of local fields, there is a reciprocity isomorphism

G(LIKtb ;:: K*/NLIKL*.

The multiplicative group K* may be interpreted in K -theory as the group KJ (K) of the field K. The group K 2(K) is defined to be the quotient group

K 2(K) = (K* ® K*)/ R,

where R is generated by all elements of the form x ® (l - x). Treating Galois extensions L I K of "2-local fields" - these are discretely valued complete fields with residue class field a local field (e.g., Qp((x)), IF'p((x))((y)) ) - the Japanese mathematician KAZUYA KATO (see [83]) has established a canonical isomorphism

G(LIK)ab ;:: K2(K)/NLIKK2(L).

Kato's proof is intricate and needs heavy machinery. It was simplified by the Russian mathematician I. FESENKO (see [36], [37], [38]). His proof may be viewed as a special case of the theory sketched above. The basic idea is the following. The correspondence K ~ K 2(K) may be extended to a G-modulation K 2. It does not satisfy the hypothesis of exercise 15, so that one may not apply the abstract theory directly to K 2. But FESENKO considers on K2 the finest topology for which the canonical map ( , ) : K* x K* -* K 2(K) is sequentially continuous, and for which one has Xn + Yn -* 0, -Xn -* 0 whenever Xn -* 0, Yn -* O. He puts

K~oP(K) = K 2(K)/A2(K),

where A2(K) is the intersection of all open neighbourhoods of 1 in K2(K), and he shows that

K~OP(K)/NLIKK~OP(L) ;:: K2(K)/NLIKK2(L)

for every Galois extension LIK, and that K~oP(K) satisfies properties which imply the hypothesis of exercise 18 and 19 when viewing K~oP as a G-modulation. This makes KATO'S theorem into a special case of the theory developed above.

§ 7. The Herbrand Quotient

The preceding section concluded abstract class field theory. In order to be able to apply it to the concrete situations encountered in number theory,

§ 7. The Herbrand Quotient 311

it is all important to verify the class field axiom (6.1) in these contexts. An excellent tool for this is the Herbrand quotient. It is a group-theoretic formalism, which we develop here for future use.

Let G be a finite cyclic group of order n, let (J' be a generator, and A a G-module. As before, we form the two groups

HO(G, A) = AG jNGA and H-l(G,A) = NGAjIGA,

where

n-l .

A G = { a E A I aU = a} , N G A = { N Ga = TI aU' I a E A} , i=O

NGA = {a E A I NGa = I} , IGA = {aU- l I a E A} .

(7.1) Proposition. If 1 ---+ A ---+ B ---+ C ---+ 1 is an exact sequence of G -modules, then we obtain an exact hexagon

Y HO(G,A)

H-1(G,C)

~ H-1(G,B) E J4

i Proof: The homomorphisms 11,14 and /z,15 are induced by A -+ B and B ~ C. We identify A with its image in B so that i becomes the inclusion. Then h is defined as follows. Let C E CG and let b E B be an element such that j(b) = c. Then we have j(bu - l ) = cu - l = 1 and NG(bU- l ) = NG(bU)jNG(b) = I, so that bu - l E NGA. h is thus defined by c mod NGC 1-+ bu - l mod IGA. In order to define 16, let c E NGC, and let b E B be an element such that j(b) = c. Then j(NGb) = NGc = I, so that NGb E A. The map 16 is now given by c mod IGA 1-+ NGb mod NGA.

We now prove exactness at the place HO(G, A). Let a E AG such that !t(a mod NGA) = 1; in other words, a = NGb for some bE B. Writing c = j(b), we find 16(C mod IGC) = a mod NGA. Exactness at H-l(G, A) is deduced as follows: let a ENG A and 14 (a mod I G A) = I, i.e., a = bU -1,

with b E B. Writing c = j(b), we find h(c mod NGC) = a mod IGA. The exactness at all other places is seen even more easily. 0


(7.2) Definition. The Herbrand quotient of the G -module A is defined to be

h(G A) = #Ho(G,A) , , #H-l(G,A)

provided that both orders are finite.

The salient property of the Herbrand quotient is its multiplicativity.

(7.3) Proposition. If 1 -+ A -+ B -+ C -+ 1 is an exact sequence of G-modules, then one has

h(G,B) = h(G,A)h(G,C)

in the sense that, whenever two of these quotients are defined, so is the third and the identity holds.

For a finite G-module A, one has h(G, A) = 1.

Proof: We consider the exact hexagon (7.1). Calling ni the order of the image of Ji, we find

#Ho(G, A) = n6nl,

#H-1(G,A) = n3n4,

and thus

#Ho(G,B) = nln2,

#H-1(G, B) = n4nS,

#Ho(G,A) .#Ho(G,C) .#H-1(G,B)

#Ho(G, C) = n2n3,

#H-1(G,C) = nSn6,

= #Ho(G, B) . #H-1(G, A) . #H-1(G, C).

At the same time, we see that if any two of the quotients are welldefined, then so is the third. And from the last equation, we obtain h(G, B) = h(G, A)h(G, C). Finally, if A is a finite G-module, then the exact sequences

show that #A = #Ao . #/0 A = #NGA· #NoA, and h(G, A) = 1. 0

If G is an arbitrary group and g a subgroup, then to any g-module B, we may associate the so-called induced G-module

A = Indb(B).


It consists of all functions I : G --+ B such that I(Xi) = I(x)' for all i E g. The operation of (1 EGis given by

la (x) = 1«(1x).

If g = {I}, we write IndG(B) instead of Ind~(B). We have a canonical g -homomorphism

Jr : Ind~ (B) --+ B, I t--+ 1(1),

which maps the g -submodule

B' = { I E Ind~ (B) I I (x) = 1 for x ¢ g} isomorphically onto B. We identify B' with B. If g is of finite index, we find

Ind~ (B) = Il B a , aEG/g

where the notation (1 E G / g signifies that (1 varies over a system of left coset representatives of G / g.

Indeed, for any I E Ind~ (B) we have a unique factorization I = Ila 1% ' where la denotes the function in B' which is determined by la(l) = 1«(1-1).

If conversely A is a G-module with a g-submodule B such that A is the direct product

then A ~ Ind~(B) via B ~ B'.

(7.4) Proposition. Let G be a finite cyclic group, g a subgroup and B a g -module. Then we have canonically

Hi(G, Ind~(B») ~ Hi(g,B) for i =0,-1.

Proof: Let A = Ind~ (B) and let R be a system of right coset representatives for G / g with I E R. We consider the g -homomorphisms

Jr : A --+ B , I t--+ 1(1) ; v : A --+ B , I t--+ Il I (p ) . pER

Both admit the g -homomorphism

s:B--+A,

as a section, i.e., Jr 0 S = v 0 s = id, and we have

Jr 0 NG = Ng 0 v,

for (1 E g,

for (1 fj g,


because one finds that, for f E A,

(Nof)(!) = n n f P' (1) = n n f(pr) = n(n f(p))' = N g( v(f)) . 'Eg pER , P , P

If f E A O , then f(a) = f(1) for all a E G, and f(1) = fer) = f(1)' for all rEg. The map n therefore induces an isomorphism

n : A O --+ Bg.

It sends NoA onto NgB, for one has n(NoA) = Ng(vA) ~ NgB on the one hand, and on the other, Ng(B) = Ng(vsB) = n(No(sB)) ~ n(NoA). Therefore HO(G, A) = HO(g, B).

As N g 0 v = n 0 No, the g-homomorphism v : A ~ B induces a g -homomorphism

v: NGA --+ NgB.

It is surjective since v 0 s = id. We show that loA is the preimage of 19B. loA consists of all elements fa-I, f E A, a E G. For if G = (ao) and

a = a~, then f a - 1 = fCl+ao+··+a6-I)(ao-J) E loA. In the same way, one

has 19B = {b,-l I b E B, rEg}. Writing now ap = p'rp , with p,p' E R,

rp E g, we obtain

v(fa-l) = n f(ap) = n f(p')'p = n bl-1 E 19B. pER f(p) pi f(p') p

On the other hand, for b,-1 E 19B, the function f,-I, with f = sb, is a preimage as v(f,-l) = vs(b),-l = b,-I. After this it remains to show ker(v) ~ loA. Let G = (<p), n = (G : g), R = {I, <p, ... , <pn-l}.

Let f ENG A be such that v(f) = n7~d <pi = 1. Define the function

h E A by h(1) = 1, h(<pk) = n~~i f(<pi). Then f(<pk) = h(<pk)j h(<pk-l) = h(<pk-l)l-rp-I for 0 < k < n, and fO)hrp-I_I(1) = n7~d f(<pi) = 1. Hence

f = h1-rp-1 E loA. Thus we finally get H-tcG,A) = H-1(g,B). 0

Exercise 1. Let f, g be endomorphisms of an abelian group A such that fog = g 0 f = O. Make sense of the following statement. The quotient

(ker f : img) q/,g(A) = (ker g : im f)

is multiplicative.

Exercise 2. Let f, g be two commuting endomorphisms of an abelian group A. Show that

qO,g/(A) = qO,g(A)qO,f(A) ,

provided all quotients are defined.


Exercise 3. Let G be a cyclic group of prime order p, and let A be a G-module such that qo,p(A) is defined. Show that

h(G, A)p-l = qo,p(AGy /qo,p(A).

Hint: Use the exact sequence

0--+ AG ---+ A ~ Au - 1 --+ O.

Let N = 1 + a + ... + a P- 1 in the group ring Z[GJ. Show that the ring Z[GJ/ZN is isomorphic to Z[n, for ~ a primitive p-th root of unity, and that in this ring one has

where E is a unit in Z[GJ/ZN.

Exercise 4. Let L I K be a cyclic extension of prime degree. Using exercise 3, compute the Herbrand quotient of the group of units o! of L, viewed as a G(L IK)-module.

Exercise 5. If G is a group, g a normal subgroup and A a g-module, then Hl(G, Ind~(A)) ;;::: Hl(g, A).

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