(Encyclopaedia of Mathematical Sciences) A. N. Parshin, I. R. Shafarevich - Number Theory 2_ Algebraic Number Theory-Springer (1992)

The Principal Ideal Theorem

Theorem 2.18 Let K be an algebraic number field and H the Hilbert class field of K Then every ideal of K becomes principal considered as ideal of H

By means of Theorem 2.10 we can reformulate Theorem 2.18 as a purely group theoretical statement: Let H’ be the Hilbert class field of H Then H’/K is

104 Chapter 2 Class Field Theory normal and H/K is the maximal abelian subextension of H’/K Therefore The- orem 2.18 is equivalent to the statement

By Theorem 2.10 (2.22) is equivalent to

Hence Theorem 2.18 follows from the group theoretical ideal theorem:

Theorem 2.19 Let G be a finite group with abelian commutator group H :[G, G] Then the transfer map from G to H is trioial lg~ (Artin, Tate (1968),

Beginning with Taussky (1932) several authors investigated the process of

“capitulation”, i.e becoming a principal ideal, of the ideals of K in the subfields of the Hilbert class field For a more general point of view and as reference source see Schmithals (1985).

Local-Global Relations

Let L be an abelian extension of the algebraic number field K, and let N denote the corresponding subgroup of the idele class group E(K) Given a place w of L that lies above the place u of K, Theorem 2.8 indicates that the norm group of N plays a significant role in understanding the relationship between these fields.

In the context of class field theory, if we have a finite set S of places of a field K where local abelian extensions L_v/K_v exist, a natural inquiry is whether there is a global abelian extension L of K This extension L should be such that its localization at the places v in S is isomorphic to the local extensions L_v/K_v The resolution to this question is framed in terms of subgroups of the class group CC(K), and a positive answer is provided by a specific theorem.

Theorem 2.20 Let P be the group nvsS KG with the topology given by the natural injection n K,” + W) VES and let F be the group nvcs K,” with the product topology

Then a subgroup V of finite index is closed in P $ and only if it is closed in P

For any such subgroup V there exists a closed subgroup N of E(K) such that

The question arises whether a global extension L/K can be established with specific local properties and the condition that the degree [L : K] equals the least common multiple of the local degrees [L : K] for each u in S Grunwald (1933) asserted that this holds true for cyclic extensions; however, Wang (1950) identified an error in Grunwald's proof and provided the correct formulation of the theorem, now known as the Grunwald-Wang theorem (Theorem 2.22 below).

5 1 The Main Theorems of Class Field Theory 105

There are “exceptional” cases in which Grunwald’s claim is wrong These cases are connected with the following theorem on m-powers in K:

Theorem 2.21 Let m be a natural number and S a finite set of places of

K Moreover let s be the greatest natural number such that K(q,) = K, where n, = izs + [,;’ and C2 is a primitive root of unity of order 2” Then the group

P(m, S) := {CX E K” ICI E K,“” for u # S} is equal to K”” except under the following conditions which will be called the special case:

1 - 1, 2 + n,, and - (2 + n,) are non-squares in K

2 m = 2fm’, where m’ is odd, and t > s

3 So E S, where So is the set of prime divisors p of 2 in K for which - 1,2 + I],, and -(2 + r],) are non-squares in K,

In the case where K equals Q, we find that s equals 2, particularly when 8/m and 2 are elements of S Given that Q equals 0, it follows that a0 equals 2mi2 Notably, 16 is recognized as an 8-th power in the context of cc and all odd primes; however, it does not qualify as an 8-th power in Q or Q2 This leads us to the formulation of the Grunwald-Wang theorem.

Theorem 2.22 states that for a finite set of places $ S $ of a field $ K $ and local characters $ x_v $ of period $ n $ for each $ u \in S $, there exists a global character $ x $ on $ C(K) $ with local restrictions matching the given $ x_v $ Furthermore, if the condition $ pgo \, x_p(a_0) = 1 $ is met, the global character can be constructed with period $ m $, which is the least common multiple of the $ n_v $ However, if this condition is not satisfied, the resulting global character can only achieve a period of $ 2m $.

When the degrees of the extensions LJK are specified locally, it is possible to bypass a particular case The theorem derived from this scenario has significant implications in the theory of simple algebras over number fields, as highlighted in Theorem 2.89.

Theorem 2.23 states that for a finite set of places S of a field K, where each place v in S is associated with a positive integer n, if v is archimedean, then n should represent a feasible degree for an extension of K Consequently, there exists a cyclic extension L/K with a degree equal to the least common multiple of the n values, ensuring that the completions L/JK have the specified degree.

The Zeta Function of an Abelian Extension

106 Chapter 2 Class Field Theory the Dedekind zeta function, and let /IL(s) be the corresponding enlarged zeta function (Chap 1.6.3)

Theorem 2.24 Let U = NLIKJ(L) be the subgroup of J(K) corresponding to L and let X be the group of characters x of J(K) which are trivial on U Then

Proof For the proof of (2.23) one has to show n (1 - %,dW “) = (1 - K,Q(P)-fS)g = n (1 - x(P)~K,,(P)-“), (2.25)

XEX where g denotes the number of prime divisors ‘$3 in L of the prime ideal p and S is the inertia degree of ‘$J in L/K The equation (2.25) is equivalent to the polynomial equation,

By Theorem 2.8 the characters of J(K)/U correspond to the characters of

The product defined in equation (2.26) is calculated over all unramified characters, which are those that remain trivial on the inertia group TP Given that f represents the order of G/TP, it follows that GQ/T9 is generated by the prime element (71, L/K).

K,, we see that there are exactly g such characters with x(p) = x(n) a given root of unity in ,u~ This proves (2.23)

For the proof of (2.24) one uses Legendre’s formula

T(s) = 2”-‘71-l’2r(s/2)r((s + 1)/2) cl (2.23) together with Theorem 1.104 implies

Theorem 2.25, known as the analytical class number formula, indicates that L(l, x) is non-zero for x not equal to x0 This finding is crucial as it establishes the existence of infinitely many prime ideals within the classes of the ray class group H, as supported by Theorem 1.111.

Furthermore the functional equation for n(s, x) (Theorem 1.104) together with

Theorem 2.26 (Product formula for the &-factors)

For the computation of+(X) we can without loss of generality assume that L/K is the cyclic extension corresponding to the kernel of x Let x # x0 be a real

0 2 Complex Multiplication 107 character, i.e x2 = x0 Then 1x1 = 2 and since E&J = 1, (2.28) implies Ed 1 Hence (1.55) determines the Gaussian sum z(x):

Theorem 2.27 Let x be a real character Then

T(X) = i-‘U%,69fXP2 where r is the number of real places in the conductor of x and f, is the finite part of the conductor of x •J

Example 9 Let K = Q and let x be a real character with conductor fu', r = 0,

1 Then Theorem 2.27 together with Proposition 1.106 imply Gauss’ formula f li2

Every abeiian extension of Q is contained in a cyclotomic field, i.e in a field which is generated by a value of the function exp(2nix) at a division point x1 of the lattice Z in Q (0 1.1)

Something similar one has for an imaginary quadratic field K as base field: The ring li, = Zo, + Zo,, Im wl/w2 > 0, of integers in K form a lattice in

In the context of complex multiplication, let K represent a field defined by a0, E D for a in DK The elliptic modular function j(z) plays a crucial role, as K(j(o, w)) denotes the Hilbert class field of K Notably, every abelian extension of K is encompassed within a field K(j(o, w)), T(z), where z(z) corresponds to Weber’s z-function, which is proportional to Weierstrass’ p-function Furthermore, z1 serves as a division point of the lattice 0.

The class field theory of imaginary quadratic number fields can be developed using the functions j(z) and z(z), as demonstrated by Deuring in 1958 This article focuses on exploring the number theoretical properties of specific values of these functions, grounded in the fundamental theorems of class field theory.

The Main Polynomial

A matrix with a determinant greater than zero is considered primitive if the greatest common divisor of its entries (a, b, c, d) is equal to one The set of such j matrices with a specific determinant, denoted as A, has orbits that are represented by triangular matrices where a is greater than zero, ad equals the determinant s, and the entries (a, b, d) are coprime with the condition that b is less than d This collection of matrices is referred to as the set B.

JS(x, j(z)) := s& (x - j(S(z))) with S(z) 108 Chapter 2 Class Field Theory has coefficients in Z[j(z)] The highest coefficient of the polynomial JS(x, x) is equal to + 1 0

The main point of the proof is to use that the Fourier expansion j(z) = q-1 + cg + Cl q + ) q = eZRiZ, has integer coefficients and that j(z) generates the field of modular functions on

Proposition 2.29 Let s = p be a prime Then

The First Main Theorem

the order in K with conductor f (Chap 1.1.1) The complete modules a with order E)f such that the ideal aDO, of K is prime to f form a group !IJIf (Chap

1.1.6) The correspondence cp: a -+ aD, defines an isomorphism of !JJ$ onto %f

(Chap 1.5.5) We put q-‘(b) = bf for b E !llr

Since there are modules of ‘%IIf in every class of CL(Dr) = 9JI(DJ)/!$0,), we have CL(E),) = !JJ,./(!JJI, n G(E),))

Let @jf be the group of principal ideals crnK such that a0, E 21f and c1 E r

In the context of class field theory, we define $ a \equiv r \mod f $ to mean that $ v(c - r) \geq v(f) $ for some $ r \in \mathbb{Q} $ This relationship allows the mapping $ a \mapsto aQK $ to induce an isomorphism from the class group $ CL(D_r) $ to the quotient $ \mathbb{uf}/\mathbb{ef} $ Consequently, $ CL(D_r) $ is understood as a class group, referred to as the ring class group mod $ J $, through this isomorphism.

Let a = a,Z + a2Z, Im(ai/cr,) > 0, be any complete module with order Df

Thenj(a) :=j(~~/c1~) depends only on the class si E CL@,) of a and determines this class We putj(a) := j(a)

Theorem 2.30 states that the set { j(3)/ si E CL(Df) } constitutes a complete collection of conjugated algebraic integers related to the field K Additionally, the field Kf, defined as K(j(a)), represents an abelian extension and serves as the class field for the ring class group CL(Dr).

To establish proof using general class field theory, it is essential to demonstrate two key points: first, that the j-invariant j(Z) for 3 in CL(Df) is an algebraic integer; and second, that for nearly all prime ideals p of K with degree 1, the module ‘pf is principal in Df if and only if p splits completely in K, as outlined in Theorem 1.118, which aligns with Weber’s definition of class field theory.

1 Let p k f, N,,,p = p and ii E CL(Df) Then

J,W), j(W;‘)) = 0 since apj’ = (a,, CQ) implies that a is generated by P a’

0 ff2 with P E A, If pJ- is a principal module, then j(ap;‘) = j(a) Hence j(a) is a root of the polynomial

J,(x, x) Proposition 2.28 implies that j(a) is an algebraic integer

Hence for any prime divisor ‘$3 of p in K(j(a), j(ap7’))

5 2 Complex Multiplication 109 j(a) = j(ap;‘)P (mod ‘p) or j(a)P = j(ap;‘) (mod ‘p) (2.29)

If ps is a principal module, then j(a)P = j(a) (mod VP) and if p is prime to the discriminant of j(a) it follows that p splits completely in K( j(a))/K

If p splits completely in K( j(a))/K, then j(a)P E j(a) (mod ‘$) and (2.13) implies j(a) = j(ap;‘) (mod ‘@)

Now we assume ‘$ kj(ti) -j(b) for arbitrary b E CL(D,) This excludes only finitely many p Then j(a) = j(api’) Hence pr is a principal module 0

The Reciprocity Law

The proof is based on the congruence j(a)N(P) z

Aa@‘) (mod P) for prime ideals p with p t f

For the proof of (2.30) one uses the function cp,(z) for S E A,, which is defined as follows:

,247~) cp,(z) := s qj where d(z) is the discriminant, i.e a non trivial parabolic modular form of weight

The function cps(z) has a significant advantage due to its well-understood theoretical structure at singular arguments, specifically at 2m = q(z), where q(z) is recognized as the Dedekind function As a result, cps(z) is also regular in the upper half-plane, Im z > 0.

Theorem 2.32 states that if b is an ideal of K that is prime to 6f, and if a1, a2, with Im(a1/a2) > 0, form a basis of a module af with order Df, then there exists a rational matrix B that transforms a1 and a2 into a basis of afgbj2, where g represents complex conjugation Consequently, B is defined as (pB(a1/a2), which is a number in Kr, and the condition (/I?) = b holds true.

If f = 1, this is the principal ideal theorem (6 1.11).

The Construction of the Ray Class Field

of a field of the form K( j(a), e2”‘8) for some a E K, Im a > 0 and /I E Q This is the special form of his conjecture which he called “seinen liebsten Jugendtraum”

It is wrong since the ray class field K(f) mod f is in general a proper extension

110 Chapter 2 Class Field Theory of K,(f) := K/(e ‘+) But G(K(f)/K,(f)) is an elementary 2-group (Hasse

(1970), I 0 10) Therefore the conjecture is true for abelian extensions of odd degree

For the generation of the full ray class field mod f” one uses the values of elliptic functions at division points

Let a = (or, wZ), Im oi/oz > 0, be an ideal of K and @(z, a) the corresponding

Weierstrass p-function For the purposes of number theory one has to multiply

@(z, a) by a factor g@) where e is the number of roots of unity in D,, i.e e = 6 if

0, = Z[(l + &3)/2], e = 4 if 0, = Z[n] and e = 2 in all other cases g’2’ := - 2’35g*(o)g&o)w;/A(w), g(4) := 2834g;(o)w;/d(w), g(6) := -2g36g3(o)o~/d(o) where as usual gz(w), g3(o) denote the Eisenstein series such that

T(Z, a) := g@)@(z, a)e12 is called Weber r-function of the ideal a The factor g’“’ is chosen such that T(Z, a) has a Fourier expansion

@ (qej2 + + t,q” + -9( 1 + 12u(l - $2 + 12 f nqnm(Un + u-” - 2) n,m=1 in q = e2ni, u = e2nWol with integral coefficients t,

Let N be a natural number Then

T N ,a > with x1, x2 E z, (xl, x2) + ((40) (mod W, is called an N-th division value of r(z, a) By evaluating r XlWl + x2w2

> a Laurent series of ql’lv one verifies that

(x~;,:~,N)=~ has rational coefficients which are integers if N is not a prime power and which become integers after multiplication by 1’ if N is a power of the prime 1 It follows that

GJ, a) = $1, v-l) for y E K, y $ a, is an algebraic number

If m and r are integral ideals of K and r is prime to m, then r(l, mr-‘) depends only on the ray class of r-i mod m In fact let r’ = Ar with 1” = 1 (mod m), then

If r-l lies in the ray class 2X, we write z(2l) := ~(1, mr-‘)

Proposition 2.33 Let 9I and 9I’ be ray classes mod m such that ‘%‘%I-’ consists of principal ideals Then ‘9I = 9I’ if and only if z(a) = $!I’) q The following theorem is called the second main theorem

Theorem 2.34 Let m be an ideal of fan and 2I a ray class mod m Then

1 K,(z@I)) is the ray class field mod m q i

I The proof is analogous to the proof of Theorem 2.30 It is based on the : following congruence:

Theorem 2.35 states that for a prime ideal $ p $ in $ K $ with degree one, which is coprime to $ 6Nmd,o $, it follows that $ z(‘?Ip-‘) \equiv ~(‘$l)~(p) \mod v $ for any prime divisor $ 'p $ of $ p $ in $ K,(z(cU), r(2lp-‘)) $ This congruence also leads to the derivation of the reciprocity law.

Theorem 2.36 Let ‘$I and 23 be ray classes mod m and 0, the ray class field

L mod m Then cu ( Q2,lK > j(23) = j(BW'), a ( Q,F ) T(B) = r(mr') El

Algebraic Theory of Complex Multiplication

The theory of complex multiplication can be algebraically developed within the framework of elliptic curves For an elliptic curve E defined over the complex numbers (C), the endomorphism ring End(E) is either isomorphic to a specific integer h or corresponds to an order in an imaginary quadratic number field When End(E) aligns with the latter scenario, it is said that the elliptic curve E exhibits complex multiplication.

The connection with the analytic theory is given by the following theorem

Theorem 2.37 states that if E is an elliptic curve defined over a field such that the endomorphism ring End(E) is isomorphic to an imaginary-quadratic number field K, then E is analytically isomorphic to the torus C/a for a certain module a with order D Additionally, for any module a with order D, the endomorphism ring End(C/a) is isomorphic to the order 0 in K Furthermore, if a and b are modules with order 0, the elliptic curves C/a and C/b are isomorphic if and only if b is equal to pa for some integer n in K.

112 Chapter 2 Class Field Theory Q 3 Cohomology of Groups 113

The invariant of E is equal to j(a) Hence E is defined over the algebraic number field Q(j(a)) By means of the reduction theory of E one obtains a beautiful proof of Theorem 2.3 1

For more details see Shimura (1971), Chap 4, or Lang (1973), Part 2.

Generalization

dental functions has already been emphasized by Hilbert in his twelfth problem

Hilbert (1976) emphasizes the significance of extending Kronecker’s theorem beyond rational numbers and imaginary quadratic fields to any algebraic field as the realm of rationality He considers this problem to be one of profound importance in the field of mathematics.

,farreaching in the theory of numbers and of functions

The natural extension of elliptic curves in the context of complex multiplication is the introduction of abelian varieties This generalization was first proposed by Shimura and Taniyama in 1961 for CM-fields, which are totally complex quadratic extensions of totally real number fields; however, it achieved only partial success as it did not encompass all abelian extensions Subsequent research led to the concept of Shimura varieties, as developed by Deligne in 1971 and further explored by Borel and Casselman in 1975, which are considered highly significant in the realms of arithmetic algebraic geometry and the theory of automorphic L-functions.

Stark (1976, 1982) proposed a different approach to Hilbert's twelfth problem, linking the units of abelian extensions to the values of Artin L-functions at s = 0, as also discussed by Tate (1984).

(Main references: Serre (1962), Part 3; Koch (1970), 5 3)

Cohomology of groups serves as an effective mathematical framework for demonstrating theorems in class field theory This concept paves the way for the advancement of Galois cohomology, which will be explored in the subsequent chapter.

This article introduces the cohomology of groups in its simplest form, focusing on factor systems For a broader understanding through the lens of derived functors, readers are encouraged to consult the classic text by Cartan and Eilenberg published in 1956.

Definition of Cohomology Groups

ring over L, a (left) G-module is a unitary (left) n-module For n 2 1 and the

G-module A we denote by K”(G, A) the set of mappings of the n-fold product of

G into A We put K’(G, A) := A and we transfer the addition in A to K”(G, A)

An element f of K”(G, A) is called an n-dimensional cochain

+ i$ (-l)‘f(xl, 2 xi-l, xixi+l, xi+2, , xn+l)

+(-l)“+lf(x,, ,x,) defines a homomorphism d = d, from K”(G, A) into Kn+l(G, A) with d,d,-, = 0 for n > 1 Hence K(G, A) := c;=o K”(G, A) with the endomorphism cF=o d, is a complex (Appendix 1.4)

A cocycle or factor system is a cochain f with df = 0, a coboundary or splitting factor system is a cochain f with f E Im d Of course every coboundary is a cocycle

We define the cohomology groups H”(G, A) of G and A as the cohomology groups of the complex K(G, A), i.e

Example 10 H’(G, A) = AG is the group of elements in A which are fixed by G •I

Example 11 Let L/K be a finite separable normal extension of fields Then L is a G(L/K)-module E Noether s generalization of Hilbert’s Satz 90 (Proposi- tion 2.6) means H’(G(L/K), L”) = (0) El

Example 12 If G acts trivial on A, then H’(G, A) = Hom(G, A) is the group of homomorphisms of G into A 0

Example 13 A two dimensional factor system defines a group extension of A with G The equivalence classes of such extensions are in one-to-one correspondence with the elements of H2(G, A) (Hall (1959) Chap 15) 0

Let X be an abelian group and MG(X) the group of mappings from G into X

We define a G-module structure for M,(X) as follows: gfb) =fkl) for g E G, f E MG(X)

Functoriality and the Long Exact Sequence

If H = G and cp = id, then [q, $1 is a homomorphism of G-modules

A morphism [q, $3 of A into B induces a morphism of K(G, A) into K(H, B) and therefore a homomorphism [q, $1, of H”(G; A) into H”(H, I?)

Example 14 Let H be a subgroup of G, B = A, II/ = id and cp the embedding of H in G Then Res := [q, $1, : H”(G, A) + H”(H, A) is called restriction from

Example 15 Let H be a normal subgroup of G Then AH is a G/H-module and we have the natural morphism [q, $1 with $: An + A, cp: G -+ G/H The corresponding homomorphism Inf: H”(G/H, AH) + H”(G, A) is called inflation from G/H to G 0

Let G and H be groups A commutative diagram

(0) -+ A’ -+ B’ + c’ + {O}, where the first row is an exact sequence of G-modules and the second row is an exact sequence of H-modules, induces an exact and commutative diagram

(0) + K(H, A’) + K(H, B’) -+ K(H, C’) + (0) of complexes Hence one has an exact and commutative diagram

(2.3 1) where A, is the n-th connecting homomorphism for the cohomology groups of the complexes The rows of (2.31) are called long exact sequences.

Dimension Shifting

of the mapping which associates to a E A the function f,(x) = xa for x E G Let

(0) + A + M,(A) -+ c -+ (0) be the corresponding exact sequence By Proposition 2.38 the associated connecting homomorphism

H”(G, C) + Hn+l(G, A) is an isomorphism for n 3 1 and a surjection for n = 0

By means of this dimension shifting it is possible to proof theorems about cohomology groups by reducing them to small dimensions This method is based on the following principle:

Proposition 2.39 states that if G and H are groups and F is an exact covariant functor mapping H-modules to G-modules while preserving induced modules, then for any integer m > 0, there exists a functorial morphism for every H-module A This morphism ensures that the induced module i(A) acts as a homomorphism from H”(G, FA) to H”(H, A).

Then there is one and only one family (&In = m, m + 1, } of functorial morphisms such that for all exact sequences of H-modules

H”+‘(G FA) 1,,10 H”+l(H, A) is commutative If I, is an isomorphism, then also II, is an isomorphism for n 2 m cl

In the following sections we give some applications of Proposition 2.39.

Shapiro’s Lemma

Take F := Mz and let i,(A) := $,,(A) be the homomorphism which is induced from the morphism [q, $1 : [G, M:(A)] -+ [H, A] where cp is the injection

It is easy to see that the assumptions of Proposition 2.39 are satisfied Since eO,(A) : (Mg(A))G = AH + AH is the identity, we have proved

Proposition 2.40 (Shapiro’s lemma) ICI,,(A) is an isomorphism q

Corestriction

we constructed a homomorphism, called restriction, from H”(G, A) into H”(H, A) Now we want to construct a homomorphism from H”(H, A) into H”(G, A), called corestriction

Take F(A) = A and let 1,(A) = Cor be given by

& is a functorial morphism and can be extended according to Proposition 2.39 to higher dimensions (observe that the roles of G and H are exchanged) The resulting morphism is the corestriction Cor

Proof This is trivial in dimension 0 and follows in higher dimension from Proposition 2.39 0

Proposition 2.42 Let G be a finite group and let A be a G-module Then H”(G, A) is annihilated by IGI for n 2 1

Let G be a finite group and H a subgroup of G We consider Q/Z as G-module with trivial action of G Then

116 Chapter 2 Class Field Theory induces a homomorphism G/[G, G] + H/[H, H], which is the transfer map defined in 0 1.6.

The Transgression and the Hochschild-Serre-Sequence

subgroup of G and A a G-module For every g E G we define an automorphism g of H”(H, A): @ is induced by the morphism [q, $1: $(a) = ga, q(h) = g-‘hg for a E A, h E H

The action of the element g allows the group H”(H, A) to be defined as a G-module Dimension shifting reveals that g acts as the identity for g in H Consequently, H”(H, A) can be viewed as a G/H-module, and the image of the restriction map H”(G, A) is obtained.

H”(H, A) lies in H”(H, A) G’H The transgression is a functorial homomorphism from H”(H, A)Gm into H”+l (G/H, AH), which is defined only if H’(H, A) = (0) for i = 1, n - 1

For n = 1 the transgression is defined as follows: let a E H’(H, A)Gm, da = 0

We extend a to an element b E K’(G, A) with the properties gb(g-‘hg) - b(h) hb(g) - b(g), b(hg) = b(h) - hb(g) for g E G, h E H This is possible since a is invariant by G Now we put f(sl, gd := db(g,> 92) for gl, g2 E G

Then f(gi, g2) depends only on the classes of gi, g2 mod H and has values in

An Hence it defines a class Tra 5 in H’(G/H, AH), the transgression

One checks directly that the following sequence is exact:

In the general case one defines the transgression and proves the following proposition by dimension shifting

Proposition 2.43 Let H be a normal subgroup of G and suppose that

H’(H, A) = (0) for i = 1, n - 1 Then the following Hochschild-Serre-sequence is exact:

One can establish the transgression and Proposition 2.43 also by means of the

Hochschild-Serre spectral sequence HP(G/H, Hq(H, A)) a H”(G, A) (Cartan-

As application we prove a proposition which will be useful in the following:

Proposition 2.44 states that for a finite group G and a G-module A, with integers n and m greater than zero, two conditions must be satisfied First, the homology groups H’(H, A) are trivial for all subgroups H of G and for all integers i in the range 0 < i < n Second, if H is a normal subgroup of H’ in G and the quotient H

Then H”(G, A) is of order dividing ICI”

In this proof, we consider a prime number $ p $ and a $ p $-Sylow group $ G $ According to Proposition 2.41, the restriction from $ H^n(G, A) $ to $ H^n(G_P, A) $ is injective on the $ p $-component of $ H^n(G, A) $ Therefore, it suffices to demonstrate the assertion specifically for $ p $-groups $ G $, where the order of $ G $ is $ p' $.

Then induction over s and application of Proposition 2.43 proves that H”(G, A) divides 1 G Im Cl

CupProduct

g(a o b) = ga o gb for g E G, a E A, b E B o is called a pairing from A x B into C

Example 16 Let A be a ring with trivial action of G Then the multiplication in A is a pairing from A x A into A 0

In the context of G-modules, the tensor product of two arbitrary G-modules A and B is defined by the equation $ da \otimes b = g \cdot a \otimes g \cdot b $ for $ g \in G $, $ a \in A $, and $ b \in B $ This establishes a pairing from the Cartesian product $ A \times B $ into the tensor product $ A \otimes B $, which is recognized as the universal pairing by the definition of the tensor product.

A pairing from A x B into C induces a bilinear map

H”(G, A) x H”(G, B) 3 H”+“‘(G, C) for every n, m 2 0, called cup product, which is defined as follows:

The bilinear map defined by i fuF:= f? establishes the cup product, which maps H”(G, A) x H”(G, B) into H”‘+“(G, C) This cup product arises from the homomorphism induced by every pairing from A x B into C, leading to a composition that connects the cup product of H”(G, A) x H”(G, B) with Hm+“(G, A Or B) and the homomorphism from H m+n(G, A gz B) to Hm+“(G, C) Consequently, it is essential to focus on the pairing from A x B into A Qa B.

The following properties of the cup product follow immediately from the definition

Proposition 2.45 Let A and B be G-modules Then c1 u fi = CL @B for

(0) -+ A + A’ + A” + (0) be an exact sequence of G-modules and let B be a G-module such that the sequence

(0) + B + B’ -+ B” -+ (0) be an exact sequence of G-modules and let A be a G-module such that the sequence

(0) -+ A & B -+ A &, B’ + A Oz B” -+ (0) is exact Then u u (AP) = (- l)m4n+.(~ u B) for ct E H”(G, A), p E H”(G, B”)

Propositions 2.46-2.47 show that the method of dimension shifting is applicable to the cup product: With

{~}~Ao,B~M,(A)o,B~co,B~{~} is exact This in particular can be applied to the proof of the following propositions

Proposition 2.48 Let G and H be groups and let [q, +,J, [q, 11/8] be morphisms of the G-modules A, B into the H-modules A’, B’ Then w,4 0 vb)*(a u D) = $3 u VGB for LX E H”(G, A), j? E H”(G, B) 0 (2.33)

(2.33) together with Propositions 2.45-2.47 determine the cup product uniquely up to isomorphy

A, B, C be G-modules with pairings (A 0 B) 0 C and

= a 0 (b 0 c) for a E A, b E B, c E C kfuP)ur = ~U(SUY) for CI E H’(G, A), a E H”(G, B), c( E H”(G, C) 0

Proposition 2.50 Let A, B be G-modules and let A o B and B 0 A be pairings such that aob=boa for a E A, b E B

9: 3 Cohomology of Groups 119 au/?=(-l)““(j?ua) for CI E H”(G, A), p E H”(G, B) 0

Proposition 2.51 Let A, B be G-modules, let H be a subgroup of finite index in

G and A 0 B a pairing Res denotes the restriction from G to H and Cor denotes the corestriction from H to G Then

Res(a u /?) = Res LX u Res b for cx E H”(G, A), fi E H”(G, B), (2.34) Cor(Res CI u fl) = c1 u Cor /I for CI E H”(G, A), b E H”(H, B) q (2.35)

Modified Cohomology for Finite Groups

In the context of G-modules, M(A) is isomorphic to Z[G] ⊗ A, where the action of G is defined as noted in (2.32) This isomorphism is established through the mapping f ↦ Σg∈G g ⊗ g⁻¹ Additionally, there exists an injection of the G-module Z, with a trivial action of G, into Z[G], represented by h ↦ Σg∈G hg for h in Z Defining J as Z[G]/Z leads to an exact sequence that captures the relationships between these structures.

(O}~A=ZO,A-rZ[G]O,A-*J,O,A~{O} implies the isomorphism

On the other hand we have a surjection ;Z[G] -+ Z given by xSEG n,g -+ c 9E G n, Let I, be its kernel Then we have the isomorphism

The group fi’(G, A) := H’(G, I, @ A) E AG/trGA is called the modified zero dimensional cohomology group

We write J,‘$ Zz for the n-fold tensor product of J,, I, respectively Then we have functorial isomorphisms

We define the modified cohomology groups for higher dimensions by fi”(G, A) := Z?‘(G, Z;” @ A) for m < 0, Z?“(G, A) := E?‘(G, JE @ A) for m > 0

The principle of dimension shifting allows us to formulate the definitions and propositions from sections 9 3.2-a 3.7 for the groups fim(G, A), m E Z, with minimal alterations We encourage the reader to explore this further.

Example 18 The elements of (I, @ A)’ have the form xse c g @ ga for a E A withC,,,ga=O.a+z 9E G g Q ga induces an isomorphism where I,A := (uaju E I,, a E A) = (ga - a/g E G, a E A) 0

Example 19 k2(G, Z) g k’(G, 1,) g Io/IeIe g G/[G, G] The isomorphism G/[G, G] + &/IGIG is given by g[G, G] + g - 1 + ICI, 0

I+(G, Z) x I?“(G, Z) + fi”(G, H) E H/IGI h defines a perfect pairing, i.e the induced homomorphism

&“(G, 22) z Hom(&(G, h), h/lGI Z) (2.36) is an isomorphism (compare Proposition 2.42) Cl

Let Q be the G-module with trivial action of G Then fi”(G, Q) = (0)

(Proposition 2.42) Therefore the exact sequence

(0) + z -+ Q + a/z + (0) induces isomorphisms A”(G, Q/H) g 8”“(G, H), n E H In particular fi2(G, Z) g fi’(G, Q/Z) g Hom(G, Q/E) Hence Proposition 2.52 implies the isomorphism k2(G, Z) g G/[G, G] of Example 19 (2.35) shows that the transfer map (0 1.6,

0 3.5) corresponds to the restriction Hm2(G, Z) + gW2(H, Z) (see Serre (1962),

Cohomology for Cyclic Groups

with generator s The isomorphism of I, onto JG given by si+l - si + si + z for i = 1, , m - 1 induces isomorphisms

S,, can also be described as follows: Let x E H’(G, Q/Z) be the character with x(s) = A + Z and let d,: fi’(G, Q/Z) + e’(G, Z) Then u-4 = u u d,(x) for CI E @(G, A) (2.37)

A G-module A such that fi’(G, A) and @‘(G, A) (and therefore @‘(G, A) for n E Z) are finite is called Herbrand module The quotient h(A) := Ifi’(G, A)//@-‘(G, A)1 is called Herbrand quotient

Let (0) -+ A + B + C + (0) be an exact sequence of G-modules, two of which are Herbrand modules The long exact sequence (2.3 1) implies that also the third

Q 4 Proof of the Main Theorems of Class Field Theory 121 module is an Herbrand module and that h(B) = h(A)h(C) (2.38)

Proposition 2.53 Let A be a finite G-module Then h(A) = 1 0

For a G-module A we denote by h,(A) the Herbrand quotient of the abelian group A with trivial action of G if it exists If A is finitely generated, then h,(A) = IGirkA

Proposition 2.54 Let p be a prime, G a group of order p, and A a G-module such that h,(A) exists Then also ho(AG) and h(A) exist and h(A) is given by h(A)P-’ = ho(AG)“/ho(A) iXl (2.39)

The Theorem of Tate

Theorem 2.55 (Theorem of Tate) Let G be a finite group and let A be a G-module such that H’(H, A) = (0) and H2(H, A) is cyclic of order IHI for euery subgroup H of G

Let c be a generator of H’(G, A) Then the homomorphism (P”: I?“(G, H) -+ I?+2(G, A) given by cp,(cr) = o! u [ for c1 E A”(G, Z) is an isomorphism for all nE27.O

$4 Proof of the Main Theorems of Class Field Theory

(Main reference: Cassels, Friihlich (1967), Chap 6, 7; Chevalley (1954))

In this paragraph we show how one can prove the main theorems of class field / theory using cohomology of groups

Application of the Theorem of Tate to Class Field Theory

1 defined the notion of a class module A, of a local or global field K From the

I point of view of cohomology the following theorem contains the fundamental facts about class modules

Theorem 2.56 states that for a finite normal extension L/K of local or global fields with Galois group G, the first cohomology group H’(G, AL) is trivial, denoted as (0) Additionally, the second cohomology group H2(G, AL) forms a cyclic group of order 1, represented as GI, and has a canonical generator known as the canonical class, denoted cLIK.

For local fields, H’(G, AL) = (0) is E Noether’s generalization of Hilbert’s Satz 90 (Proposition 2.6) All other statements of Theorem 2.56 are deep results,

I being an essential part of class field theory The proof of Theorem 2.56 will occupy us in the following sections

Combining Theorem 2.55 with Theorem 2.56 we get

Theorem 2.57 Let L/K be a finite normal extension of local or global fields with Galois group G Then there is a canonical isomorphism between the groups / / I?“(G, Z) and I?“+‘(G, A,) for any n E Z

In particular for n = -2 we have

This demonstrates a key aspect of the main theorem in class field theory, applicable in both local and global contexts Notably, the homomorphism from A to G/[G, G], as derived from equation (2.40), corresponds to the norm symbol referenced in section 1.3, § 1.5.

The canonical class induces a monomorphism inv: H2(G, AL) + Q/Z, called the invariant, which is defined by inv cL,K = ~ 1

One has the following characterization of the norm symbol: If x E H’(G, Q/Z) and d, is the connecting homomorphism H’(G, Q/Z) + H2(G, Z), then x(a, L/K) = inv(a u d 1 x) (Serre (1962), Chap 11.3) for a E A, (2.41)

The behavior of the cohomological operations deliver us the functorial properties of the norm symbol (9 1.6)

Since we consider in this paragraph arbitrary normal extensions L/K we get something more than the results stated in 4 1 For example one has the following theorem about the norm:

Theorem 2.58 Let LfK be a finite extension of local or global fields and let

L’jK be the maximal abelian subextension of L/K Then N,,,A, = NLSIKAL9

Proof Let E be a normal extension of K containing L We put G :G(E/K), H := G(E/L), then G(E/L’) = [G, G]H Let a E N,.,K(AL.) and therefore

Given that (a, L//K) equals 1, it follows that (a, E/K) is part of the image of Ha* + Gab Since A, + Ha* is surjective, equation (2.4) indicates the existence of an element CI in A such that (NLIKa, F/K) corresponds to (a, F/K) Consequently, from equation (2.40), there exists another element CI’ in A such that NE,& NLIKa/a, leading to the conclusion that a can be expressed as N,&cr/N,,,a’).

Class Formations

In class field theory, we typically present class modules as axioms and derive key theorems from them While this is often done in an abstract way, our focus will be specifically on field formations.

Let F represent an arbitrary field, and let F be its separable algebraic closure A field formation over F is defined as a covariant functor A that maps the category R of finite extensions K/F, where K is contained in F, into the category of abelian topological groups This functor must possess specific properties that characterize its behavior within this mathematical framework.

Let K and L be fields with K contained in L The homomorphism z: A1 → A2 associated with the injection K → L is an immersion, meaning that it is injective and its image is closed in A2 We will treat A1 as a subgroup of A2 If L/K is a normal extension with Galois group G, then we define A3 as A1.

The class modules for local and global fields establish a field formation over Q, and these are the sole field formations relevant to this article.

Q 4 Proof of the Main Theorems of Class Field Theory 123

In the context of a normal extension L/K, where L and K are fields in R, we consider an intermediate field K’ We define the notation H”(L/K) as H”(G(L/K), AL), and we denote the restriction from G(L/K) to G(L/K’) as Res,,,, Additionally, we apply similar notation for the inflation and corestriction maps.

A field formation A over F is called a class formation if the following axioms are fulfilled:

1 Let L/K be a normal extension of prime degree of fields in R, Then H’(L/K) = {O} and the order of H2(L/K) divides [L : K]

2 For every natural number m and any field K E R, there exists a finite abelian extension E E R, with m( [E : K] and H2(E/K) is a cyclic group of order [E : K] with a canonical generator cEIK We call EfK a special extension

3 If EjK is a special extension and K’ E R, is an arbitrary extension of K, then EK’/K’ is a special extension and the homomorphism H2(E/K) -+ H2(EK’/K’) induced from the projection G(EK’/K’) + G(E/K) maps tEIK onto [EK’ : E][,,,,,,

4 If EjK and E’/K are special extensions and v and v’ are natural numbers with [E : K]/[E’ : K] = v/v’, then v Inf,,,,, IE,K = v’ I&,,, IEcIK

We shall see in the next sections that unramified extensions in the local case and cyclotomic extensions in the global case serve as special extensions

In the rest of this section we show that our axioms allow us to give a full picture of the structure of H’(L/K) and H2(L/K) for arbitrary normal extensions L/K in si,

Proposition 2.59 Let L/K be a normal extension in si, Then H’(L/K) = (0) and the order of H2(L/K) divides [L : K]

Proof This follows from axiom 1 and Proposition 2.44 0

Proposition 2.60 Let L/K and L’/K be normal extensions in A, with L’ s L Then Inf,.,,: H’(L’/K) + H2(L/K) is injective

Proof This follows from Proposition 2.59 and Proposition 2.43 Cl

We are going now to construct a canonical generator of H2(L/K) for arbitrary normal extensions L/K in R,:

Let E/K be a special extension with [L : K]l[E : K] We put v = [E : K]/ [L: K] Then [EL : Eli,,,, = 0 since the order of cELiL is [EL : L] and v[EL : E] = [EL : L] Therefore axiom 3 implies

Hence the exact sequence to)- H2 (L/K) ~nf H2(EL/K) z H2(-WL) (Proposition 2.43) shows that there is a unique element cLja in H2(L/K) such that

Since inflation is injective, iLjg has order [L : K] It follows easily from axiom 4 that cLiK does not depend on the choice of the special extension E

Proposition 2.61 Let L/K be a normal extension in R, and let K’ be an intermediate field of L/K Then H’(L/K) is cyclic of order [L : K] with the canonical generator iLjrc Moreover

Car,,+, ILIKe = CK’ : Kli,,, (2.43) and

I&,-, IKfIK = CL : K’IL,, if K’/K is normal (2.44)

Proof The first assertion follows from the construction of iLjx and Proposi- tion 2.59, (2.42) is proved by means of axiom 3, (2.43) follows from (2.42) and

Proposition 2.41, and (2.44) follows from the construction of lLjrc Cl

The use of special extensions in the construction of the canonical generator of

H’(L/K) is called in German Durchkreuzung This method was discovered by

Chebotarev, who applied it for the proof of his density theorem (Chap 1.6.7)

Artin later utilized this method in his proof of the reciprocity law, demonstrating that the norm symbol is defined by specific properties This foundational concept serves as the basis for Neukirch's approach to class field theory, as outlined in his 1986 work.

Let L/K be a normal extension in R, and g an isomorphism of L onto gL

Then g induces the morphism [q, $1: [G(L/K), AL] + [G(gL/gK), A,,] given by $(a) = ga, q(h) = 9-l hg for a E A,, h E G(gL/gK) The resulting homomorphism from H2(L/K) to H’(gL/gK) will be denoted by g*

If we knew that the class modules for local and global fields form a class formation, for the proof of the main theorems of class field theory (Theorem 2.3,

2.8) it remained to prove that d is surjective In the scope of class formation this can be formulated as follows:

Axiom 5 For every field K E R, and every closed subgroup U of finite index in A, there exists an abelian extension L E si, of K such that NLIKAL = U

We have also to show that the norm symbol constructed in 9 1 by means of

Lubin-Tate extensions and a local-global principle coincides with the map given by (2.40) The functorial properties of the norm symbol (5 1.7) then follow from

Cohomology of Local Fields

of rational p-adic numbers We want to show that A, = K x defines a class formation over Q,

Proposition 2.63 Let L/K be a cyclic extension of local fields of degree n Then the order of H2(L/K) is equal to n

5 4 Proof of the Main Theorems of Class Field Theory 125

Using the exponential function, a submodule I/ of finite index is constructed within the group of units U, leading to the conclusion that H”(G(L/K), V) equals (0) for m in Z Consequently, the Herbrand quotient h(U,) is determined to be l, and h(L”) is equal to n Therefore, the absolute value of IH’(L/K)I is n, given that IH’(L/K)I equals 1, as stated in Proposition 2.6.

Unramified extensions of a field K are considered special extensions, characterized by their cyclic nature For every natural number n, there exists a unique unramified extension of K with degree n, as established in Chapter 1.4.6.

Proposition 2.64 Let EfK be a unramifed extension of degree n Then I?~(G(E/K), U,) = (0) for m E z

Proof Since n = IH’(E/K)I = Ifi’(E/K)I = [K” : N,,,E”] and NEigEX NEIKUEX(z”) for a prime element 7~ of K, we have NEIKUE = U, hence H’(G(E/K), U,) = (0) N ow h(U,) = 1 implies H’(G(E/K), U,) = (0) 0 The exact sequence

(l}+U,+E”-$+{O} shows together with Proposition 2.64 that vz: H2(E/K) + H2(G(E/K), Z) is an isomorphism Furthermore A,: H’(G(E/K), Q/Z) + H’(G(E/K), Z) is an isomorphism Let x be the character of G(E/K) such that x(F) = L+ Z where F n denotes the Frobenius automorphism of E/K

Now we define the canonical class iEjX by means of the equation

It is not difficult to verify the axioms 2-4 and to show that the homomorphism of K” onto G(E/K) given by (2.40) maps 71 onto F, i.e coincides with the norm symbol

One can use Lubin-Tate theory to show that axiom 5 is fulfilled (Serre (1967), 3.7-3.8).

Cohomology of Ideles and Idele Classes

Let K represent an algebraic number field and L a finite normal extension of K, characterized by its Galois group G The places of K are denoted by v, while those of L are indicated by w The decomposition group G, associated with w in relation to L/K, corresponds to the image of G(L/K) in G when w lies above v For each place v of K, we select a corresponding place w of L The complete set of places of K is referred to as PK.

G acts on n,,,” L, (5 1.4) We define an isomorphism cp from n,,,, L, onto M:-4LJ by

126 Chapter 2 Class Field Theory cp !pw ( >

Proposition 2.65 E?“(G, n,,,,Li) E g”(G,,,“, L,Q I?“(G, fl,,,,, U,) E fi”(G,,,“, U,,,) for all m E Z where U,,, is the group of units of L, Cl

For the proof of the following proposition we need a principle which will be considered in much more general form in Chap 3.1.9:

LetA, c A, E be submodules of a G-module A such that U ?I Ai = A and suppose that Ais Ai+l induces injections E?“(G, Ai) + Z?‘(G, II,+~) so that we can consider H”(G, Ai) as a subgroup of Z?“(G, I&+~), i = 1, 2, Then iirn(G, A) = U~l I?“(G, Ai)

Proposition 2.66 E?“(G, J(L)) E xVEPKgm(GWu, L;“) for all m E Z

Proof Let S be a finite set of places of K and

JSM = Js = “rIS g G “g 9 KJ Then

J(L) = u Js s and Proposition 2.64 together with Proposition 2.65 imply fi”(G, J,) E 1 H”(G,“, L;“) vos (2.45) if S contains all ramified places

Now we consider (2.45) for a sequence Si, i = 1, 2, , with ugI q = PK The above principle proves the assertion q

According to 9 4.3, 9 4.4 for every place v of K we have a canonical isomorphism H2(G,, LG) + iZ!jZ defined by iL,,x, -+ - + Z where n, := [L, : K,]

Since H’(G,, Lc) = {Oi, we have

We come now to the main problem of this section, the proof of axiom 1 for the formation module E(K) = J(K)/K’ We need some preparations

Theorem 2.68 Let L/K be a cyclic extension of prime degree p Then h@(L)) P

Let S represent a finite collection of places in K, encompassing both infinite places and those that are ramified in L, along with a selection of places that generate the ideal class group of L through their corresponding places in L We define T as the set of places in L that are situated above the places in S.

5 4 Proof of the Main Theorems of Class Field Theory 127

Since h(J,) = fl vss n, ((2.45), Proposition 2.63), it remains to show

UES where L, = Js n Lx is the group of T-units in L (Chap 1.3.8)

L, is a finitely generated group of rank 1 TI - 1 and L$ = KS is the group of S-units in K which has rank ISI - 1 (Theorem 1.58) Hence Proposition 2.53 implies h(L,)P-’ = p(lSI-l)P/plTi-l = gs ng-l/p”-‘ 0

As a corollary of Theorem 2.68 we get immediately

Theorem 2.69 (First inequality) Let L/K be a cyclic extension of prime degree p Then

The next theorem is a consequence of Theorem 1.110 We mention it here because we wish to give a purely algebraic proof of the main theorem

Theorem 2.70 Let L/K be a cyclic extension of prime degree There are infinitely many places of K which do not split in L

In this proof, we define the set S as the collection of all places of K that do not split in L, assuming S is finite We take an arbitrary idele 714 in J(K) and find an element x in K such that x - 1 a is also in K for each u in S, as stated in Proposition 1.68 Consequently, x - 1 a belongs to N,,KJ(L).

K ‘NLJKJ(L) = J(K) in contradiction to Theorem 2.69 q

Theorem 2.71 states that for a prime $ p $ within a field $ K $, if $ S $ is a collection of places in $ K $ that includes all infinite places, all prime divisors of $ p $, and a set of places that generate the ideal class group of $ K $, then a subset $ S' $ of places exists such that $ S' \cap S $ is non-empty Additionally, the natural map from this subset is surjective.

Then any x E K with x E J(K)Pnu,s U,nvES,K,X is a p-th power in K” Proof Let K’ = K(G) We have to prove K’ = K

VES ues v@ US by local class field theory (9 4.3) Since

KS + “g, WJ,P = Js(K)D is surjective, J(K) = K ‘J,(K) = Kx D hence J(K) = Kx NKfIK(J(K’)) This implies K’ = K by Theorem 2.69 Cl

We look now at the special case of Theorem 2.71 that S’ is empty We put

H := J(K)P fl U,, N := ISI v$S Then KS n H = Kg hence

[J,(K) : H] = “vs p2/pvCp) = p2N by Proposition 1.73, (1.26) ( u is here to be understood as the normalized valua- tion) Therefore

After this preparations we are able to prove axiom 1 for the field formation

Theorem 2.72 Let L/K be a cyclic extension of prime degree p Then

Proof By means of Theorem 2.68 it is sufficient to prove

(2.47) is called the second inequality

Using cohomological techniques, the equation (2.47) can be simplified to the scenario where K includes the p-th roots of unity Define L as K(s) and consider a finite set S of places in K that meet the criteria outlined in Theorem 2.71, with the additional requirement that S includes all places IJ where a is not a unit in K.

The extension Es is defined as K(&lx E KS) with a degree of pN Let g1, , gN represent a generator system of G(E/K), where L is the fixed field of gN, and we denote the fixed field of gi as Ei According to Theorem 2.70, there exists a place wi of Li that does not split in E, and we define ui as the corresponding place of K with wilt+ The mapping is surjective, and Ei serves as the decomposition field of vi in Es/K If x is in Ker cp, then vi splits in K(&)/K, leading to the conclusion that K(G) is contained in Ei for i = 1, , N, which further implies that K(fi) equals K Consequently, we find that pN equals the degree [KS: Kbl LO CK, : uo”il, confirming that pN is indeed pN.

We apply Theorem 2.71 to the set S’ = {ul, , UN) and get

Q 4 Proof of the Main Theorems of Class Field Theory 129

KXH fi Koi:Kx~ 1 =pn for n = 1, , N (2.48) i=l

Now we are looking for ideles which are in NLIK(J(L)) We find

JWP “g U, fl K; = H n Kv: E &.,K(JW i=l i=l and therefore by means of (2.48)

Analytical Proof of the Second Inequality

In this section, we demonstrate a swift proof of equation (2.47) using the techniques outlined in Chapter 1.6.6 We focus on functions of the real variable s, specifically for values where s > 1 The functions gl(s), g2(s), and so on, represent those that remain bounded as s approaches infinity.

Let L/K be a normal extension of degree n We want to show

By local consideration it is easy to see that there is a defining module m such that Kx N&J(L)) c Kx U,,, (Chap 1.55) Hence there is a subgroup 23, of 2I, containing the ray group R, such that

In the context of number theory, the set of prime ideals in 2I, denoted as C, completely splits in the extension L/K This relationship leads to the formulation of the equation clog L(s> xl = P&n: %,I x ,A m WT” + g,(s) 2 C&n: Bml & NW” + g,(s), where x encompasses all characters of the group %!l,,,/23,.

Since log L(s, x) is bounded for s + 1 if x # x0 and log w, x0) = -lo& - 1) + 92(s) we have

On the other hand we have log L,(s) = c NR-Y-” + 4) = -lo& - 1) + g&J

(2.51) where the sum runs over all prime ideals ‘$I of L which split completely in L/K (compare Theorem 1.112)

130 Chapter 2 Class Field Theory Q 5 Simple Algebras 131

We divide by -log(s - 1) Then for s + 1 one gets

The Canonical Class for Global Extensions

2-4 for RQ A cyclotomic extension of K is a subextension of some extension

In this article, we explore the cyclic cyclotomic extensions of a field K, where [ represents a root of unity We focus on constructing the canonical class for these extensions, outlining the necessary steps to achieve this goal.

For an arbitrary normal extension L/K with Galois group G we define the symbol (a, x) E H2(L/K) for a E C(K) = H’(L/K), x E H’(G, Q/Z) by

(a, x) := au A,x where A, is the connecting homomorphism H1(G, Q/Z) + H2(G, Z) It is easy to show that (a, x) has the following degeneracy properties:

= 0 for all x E H1(G, Q/Z) if and only if a E n N,,&(M)

M where M runs over all cyclic subextensions of L/K

(4 x> = 0 for all a E 6(K) if and only if x = x0

We utilize our extensive knowledge of cyclotomic fields to define the group G(Q([)/Q) for a primitive m-th root of unity, denoted as c For an element a in J(Q), we establish (O([)/Q, a) by identifying U in (Z/mZ)’ as the class corresponding to a through the isomorphism cp: J(Q)/Qx U,, -+ (Z/mZ)x This construction is outlined in Chapter 1.5.6, where Z in E(Q) has a unique representative n up.

U, = R, nPfi”ite UP Furthermore there is one and only one U E (Z/mE)” such that y up = u (mod U,,,,)

For an arbitrary algebraic number field K and a E J(K) one shows that

PWYQ k,Q a is in the image of the injection 1 G(K(c)/K) -+ G(Q([)/Q) One defines (K([)/K, a) as the preimage of (Q({)/Q, NKIQa) If E/K is an arbitrary cyclotomic extension, (E/K, a) is defined as the restriction of (K([)/K, a) for some

[ with E c K(l) Then (E/K, a) does not depend on the choice of [

Theorem 2.73 Let E/K be a cyclic cyclotomic extension Then Fi -+ (E/K, a) is an epimorphism of 6(K) onto G(E/K) with kernel NE,&(E) q

In a cyclic cyclotomic extension E/K of degree n with Galois group G, we define the canonical class of E/K by considering a generator s of G and an element x in H’(G, Q/Z) where x(s) equals ! + Z Additionally, we take an element a from J(K) such that (E/K, a) corresponds to s Consequently, we define i E,K as (6 x>.

It is easy to see that cEIK does not depend on the choice of s and a

It remains to show that cEjg satisfies the axioms 2-4 (Chevalley (1954),

In a normal extension L/K within the context of K Q, the canonical class ILIK is established as per equation 4.2, resulting in a symbol (a, L/K) belonging to G(L/K) for any element a in J(K) Additionally, it can be demonstrated that for cyclotomic extensions E/K, the relationship (a, E/K) holds true in conjunction with UK, as outlined in equation 4.

Let $ v $ be a place of $ K $ and $ w $ a place of $ L $ above $ u $ The decomposition group $ G = G(L/K) $ is associated with $ w $, and $ K $ is the decomposition field of $ w $ We define $ \tilde{w} $ as the map from $ H^2(L/K) $ to $ H^1(L/K) $, which is induced by the injection $ L^G \to C(L) $ Since the definitions of local and global canonical classes correspond, we have $ k_{SL,K} = I_{L/K} $.

This shows that (a, L/K) is the norm symbol defined in 0 1.5

To complete the proof of the main theorem in global class field theory, it is essential to establish axiom 5 This is achieved by reducing the problem to the scenario where the closed subgroup U of finite index in 6(K) possesses the characteristic that c(K)/U has a prime exponent p.

K contains the p-th roots of unity Then one uses (2.46): On one hand every U

U, for some S On the other hand the exten- field of the group J(K)/HK’:

The relationship between simple algebras over local and global fields and class field theory is profound, as cohomological methods have evolved from the application of simple algebras in this theory Additionally, the characteristics of simple algebras can be analyzed and understood through the principles established in class field theory.

Simple Algebras over Arbitrary Fields

An algebra A over K is simple if it has only the trivial twosided ideals (0) and

A With A and B also A OK B is a simple algebra over K Let A be a simple algebra over K and L a field extension of K Then A OK L is a simple algebra over L

Theorem 2.74 Let D be a division algebra over K and n a natural number Then the matrice algebra M,,(D) is a simple algebra over K An arbitrary simple algebra

A over K is isomorphic to M,,(D) for some division algebra D and some natural number n For a given simple algebra A the number n is uniquely determined and D is uniquely determined up to isomorphism H

For an algebra A over K the opposite algebra A’ is the algebra which coincides with A as vector space over K and which has multiplication (x, y) -+ yx for x, y E A

Proposition 2.75 The dimension of a simple algebra A over K is equal to n2 for some natural n

Proof Let L be the algebraic closure of K Then dim, A OK L = dim, A

Since L is algebraically closed a division algebra over L is equal to L Therefore

A OK L = M,,(L) for some number n by Theorem 2.74 Cl

Proposition 2.76 Let A be a simple algebra over K and cp: A OK A’ + End,(A) the linear map which sends a @ b to the endomorphism x + axb Then cp is an isomorphism of algebras q

Proposition 2.77 Let A be a simple algebra over K and let CI be an automorphism of A which fixes K Then a(x) = a-‘xa for some a E A

Let B, C be subalgebras of A with centers containing K and let B be an isomorphism of B onto C Then p extends to an automorphism of A H

The Reduced Trace and Norm

A a simple algebra over K Furthermore let 40 be an algebra isomorphism of A into M,(L) Such isomorphisms always exist (see the proof of Proposition 2.75)

Proposition 2.78 states that the functions z(a) = tr(cp(a)) and v(a) = det(cp(a)), where a belongs to set A, yield values in field K and are unaffected by the selection of cp Here, z(a) is referred to as the reduced trace, while v(a) is known as the reduced norm of a.

T(a + b) = z(a) + z(b), z(aa) = crz(a), z(ab) = z(ba), v(ab) = v(a)v(b) for a, b E A, cc E K

Let A be a simple algebra of degree n2 over K and L c A a field extension of

K of degree m Then m divides n and z(a) = t tr,,, a, v(a) = (NL,Ka)n/m for a E L

Splitting Fields

of K is a splitting field of A

Proposition 2.79 For every simple algebra A over K there exists a separable normal splitting field of finite degree over K Cl

Proposition 2.80 Every maximal subfield of a division algebra D over K is a splitting field of D Every splitting field of D is a splitting field of M,,(D) kl

Simple Algebras over Local Fields

The relationship between simple algebras over local and global fields and class field theory is significant, as the cohomological method has evolved from the earlier applications of simple algebras within class field theory Additionally, the characteristics of simple algebras in these contexts can be analyzed and understood through the principles established by class field theory.

In this section, we explore simple algebras over arbitrary fields, specifically focusing on algebras A that are finite-dimensional and central over a field K These algebras are defined as finite-dimensional vector spaces over K that also function as rings with a unit element, where K serves as the center of A.