We shall now take the first steps in class field theory. Specifically, since we are interested in groups of the form {J_k/k^* N J_L}, we will need their orders. And the first place to begin is with a local analog.

1. The cohomology of the units

Theorem 1 Let {L/K} be a cyclic extension of local fields of degree {n} and ramification {e}. Then {Q(U_L)=1}.


Let {G} be the Galois group. We will start by showing that {Q(U_L)=1}.

Indeed, first of all let us choose a normal basis of {L/K}, i.e. a basis {(x_\sigma)_{\sigma \in G}} such that {\tau x_{\sigma} = x_{\sigma \tau}} for all {\tau, \sigma \in G}. It is known (the normal basis theorem) that this is possible. By multiplying by a high power of a uniformizer, we find that there is a {G}-submodule {V_a} of the additive group {\mathcal{O}_L} isomorphic to {\mathcal{O}_K[G]}, i.e. is induced. We see that {V_a} has trivial Tate cohomology and Herbrand quotient 1 by Shapiro’s lemma: any {G}-module induced from the subgroup 1 satisfies this, because {H_T^i(1, A)= 0} for any {A}.

But if {V_a} is taken sufficiently close to zero, then there is a {G}-equivariant map {\exp: V_a \rightarrow U_L}, defined via

\displaystyle \exp(x) = \sum_k \frac{x^k}{k!}

which converges appropriately at sufficiently small {x}. (Proof omitted, but standard. Note that {k! \rightarrow 0} in the nonarchimedean case though!) In other words, the additive and multiplicative groups are locally isomorphic. This map (for {V_a} sufficiently small) is an injection, the inverse being given by the logarithm power series. Its image is an open subgroup {V} of the units, and since the units are compact, of finit index. So we have

\displaystyle 1 = Q(V_a) = Q(V) = Q(U_L).

This proves the theorem. (more…)