We shall now take the first steps in class field theory. Specifically, since we are interested in groups of the form , we will need their orders. And the first place to begin is with a local analog.

**1. The cohomology of the units **

Theorem 1Let be a cyclic extension of local fields of degree and ramification . Then .

Let be the Galois group. We will start by showing that .

Indeed, first of all let us choose a normal basis of , i.e. a basis such that for all . 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 -submodule of the **additive group** isomorphic to , i.e. is induced. We see that has trivial Tate cohomology and Herbrand quotient 1 by Shapiro’s lemma: any -module induced from the subgroup 1 satisfies this, because for any .

But if is taken sufficiently close to zero, then there is a -equivariant map , defined via

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

This proves the theorem. (more…)