Class Formation - The Second Inequality

The Second Inequality

The second inequality of class field theory states that

|H0(E/F)| ≤ |E/F|

for all normal layers E/F.

For local fields, this inequality follows easily from Hilbert's theorem 90 together with the first inequality and some basic properties of group cohomology.

The second inequality was first proved for global fields by Weber using properties of the L series of number fields, as follows. Suppose that the layer E/F corresponds to an extension k⊂K of global fields. By studying the Dedekind zeta function of K one shows that the degree 1 primes of K have Dirichlet density given by the order of the pole at s=1, which is 1 (When K is the rationals, this is essentially Euler's proof that there are infinitely many primes using the pole at s=1 of the Riemann zeta function.) As each prime in k that is a norm is the product of deg(K/k)= |E/F| distinct degree 1 primes of K, this shows that the set of primes of k that are norms has density 1/|E/F|. On the other hand, by studying Dirichlet L-series of characters of the group H0(E/F), one shows that the Dirichlet density of primes of k representing the trivial element of this group has density 1/|H0(E/F)|. (This part of the proof is a generalization of Dirichlet's proof that there are infinitely many primes in arithmetic progressions.) But a prime represents a trivial element of the group H0(E/F) if it is equal to a norm modulo principal ideals, so this set is at least as dense as the set of primes that are norms. So

1/|H0(E/F)| ≥ 1/|E/F|

which is the second inequality.

In 1940 Chevalley found a purely algebraic proof of the second inequality, but it is longer and harder than Weber's original proof. Before about 1950, the second inequality was known as the first inequality; the name was changed because Chevalley's algebraic proof of it uses the first inequality.

Takagi defined a class field to be one where equality holds in the second inequality. By the Artin isomorphism below, H0(E/F) is isomorphic to the abelianization of E/F, so equality in the second inequality holds exactly for abelian extensions, and class fields are the same as abelian extensions.

The first and second inequalities can be combined as follows. For cyclic layers, the two inequalities together prove that

H1(E/F)|E/F| = H0(E/F) ≤ |E/F|

so

H0(E/F) = |E/F|

and

H1(E/F) = 1.

Now a basic theorem about cohomology groups shows that since H1(E/F) = 1 for all cyclic layers, we have

H1(E/F) = 1

for all normal layers (so in particular the formation is a field formation). This proof that H1(E/F) is always trivial is rather roundabout; no "direct" proof of it (whatever this means) for global fields is known. (For local fields the vanishing of H1(E/F) is just Hilbert's theorem 90.)

For cyclic group, H0 is the same as H2, so H2(E/F) = |E/F| for all cyclic layers. Another theorem of group cohomology shows that since H1(E/F) = 1 for all normal layers and H2(E/F) ≤ |E/F| for all cyclic layers, we have

H2(E/F)≤ |E/F|

for all normal layers. (In fact, equality holds for all normal layers, but this takes more work; see the next section.)