We will spend the entirety of Chapter 4 establishing local class field theory, a classification of the abelian extensions of a local field. This will serve two purposes. On one hand, the results of local class field theory can be used to assist in the proofs of the global theorems, as we saw with Kronecker-Weber. On the other hand, they also give us a model set of proofs which we will attempt to emulate in the global case.
Recall that the term local field refers to a finite extension either of the field of \(p\)-adic numbers \(\QQ_p\) or of the field of power series \(\FF_q((t))\text{.}\) I’m going to abuse language and ignore the second case, although all but a few things I’ll say go through in the second case, and I’ll try to flag those when they come up. (One big one: a lot of extensions have to be assumed to be separable for things to work right.)
