We are now (finally) ready to start handling the cohomology of the idele classes. The previous few posts in this series contain important background computations of the Herbrand quotients of local fields and units, which should be read before this. Let be a finite cyclic extension of global fields of degree . In the following, the Herbrand quotient will always be respect to the Galois group .

Theorem 1We have . In particular,

**1. Some remarks **

The point of class field theory, of course, is that there is exactly an equality in the above statement, which is induced by an isomorphism between the two groups, and which holds for an arbitrary abelian extension of number fields.

Before we prove this theorem, let’s review a little. We know that acts on the ideles , and also on (clearly). As a result, we get an action on the **idele classes** . There is a map ; I claim that it is an injection, and the fixed points of in are precisely the points of . This can be proved using group cohomology. We have an exact sequence , and consequently one has a long exact sequence

by Hilbert’s Theorem 90, and where is the ordinary (non-Tate) cohomology groups, that is to say just the -stable points. Since we know that and , we find that , q.e.d.

So, anyhow, this Big Theorem today computes the Herbrand quotient . It in particular implies that , and since this group is none other than

we get the other claim of the theorem. We are reduced to computing this messy Herbrand quotient, and it will use all the tools that we have developed up to now.