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…)