# Examples for abstract class field theory?

I'm starting to get into Abstract Class Field Theory, following Neukirch's famous ANT.

The initial setup is basically a profinite group $G$ and a discrete abelian group $A$ on which $G$ acting as automorphisms of $A$ and such that the action is continuous.

Then there is some terminology associated with it and basic results, but before moving further on, I would like to compute a few examples to get a feel for these new objects.

Are there nice examples of such actions? Say I want to play with the profinite group $\mathbb{Z}_{p}$ (the $p$-adic integers) and the discrete abelian group $\mathbb{F}_{p}^{\times}$. Is there some interesting action as above that I can use to practice the concepts introduced? Identity action seems a little boring.

EDIT

What I would love is for example to illustrate the general reciprocity law and really get used to seeing it working in various examples before jumping into the abstract proofs.

The main goal of class field theory is to characterize Galois groups of abelian extensions of number fields (or of function fields in characteristic p), so your question about general profinite groups acting on discrete modules is not relevant, unless it concerns e.g. the cohomological approach to CFT, which starts from this kind of action . Anyway, if you ask for "nice actions" of Z${_p}$ or F*${_p}$ of an arithmetic nature, here is a rather classical exercise : fix an odd prime p and denote by $W_{p^n}$ the group of ${p^n}$-th roots of 1, $W$ the union of all the $W_{p^n}$'s. Let k = Q($W_{p}$) and K = Q($W$), g = Gal(k/Q) and G = Gal(K/k). Show that g = F*${_p}$ , G = Z${_p}$ , Gal(K/Q) = g x G. Describe the action of Gal(K/Q) on W (hint : this action is given by a homorphism of Gal(K/Q) into Z${_p}$ called the "cyclotomic character) .
• I'm sorry, what are these ${\ast}$ you used a lot of times? Mar 23, 2016 at 2:44