The above title is in fact a special case of what I want to ask.
Certainly we have a well defined notion of Galois extension for $ \mathbb{Q}_p $. The intersections of these extensions to the ring of integer of the absolute algebraic closure of $\mathbb{Q}_p$ give us a notion of Galois extensions for $\mathbb{Z}_p $. ( I know that there is a notion of Galois extension for commutative rings, and I believe that it should give us this. Am I correct?)
Let's go further. Let $A_K$ be the ring of integer in a finite Galois extension $K$ of $ \mathbb{Q}_p$. Let $e$ be the ramification degree of $K$ over $\mathbb{Q}_p$. The injection of $ \mathbb{Z}_p$ into $A_K$ will induce an injection of $ \mathbb{Z} / p^n $ into $ A_K / \mathfrak{p}^{en} $. In this picture, there seems to be some desire to say that $ A_K / \mathfrak{p}^{en} $ is the correct notion Galois of extension of $ \mathbb{Z} / p^n $. But there are problems; taking this notion of Galois extension, if $K$ is has ramification degree $e >1$, the corresponding extension $ A_K /p^e $ is not a field (it is not even an integral domain).
Question 1: Is there any notion of Galois extensions corresponding to what I desire?
Question 2: Can a class field theory (i.e a nice description of absolute abelian Galois extension) of $ \mathbb{Z}/p^n$ be developed in this context? Is there any relationship between this and the local class field theory of $\mathbb{Q}_p$ ( which is the same as that of $\mathbb{Z}_p $)?
Best Answer
Perhaps not directly answering your questions but something along those lines is Deligne's theory of truncated valuation rings, given in Les corps locaux de caractéristique $p$, limites de corps locaux de caractéristique 0.
A truncated valuation ring is an Artin local ring with principal maximal ideal and finite residue field - by Cohen's structure theorem, these are precisely the quotients of rings of integers in local fields by a power of the maximal ideal. Deligne sets up a category using these truncated valuation rings and provides definitions of extensions aswell as a ramification theory for them.
He goes on to show an equivalence between the category of "at most $e$-ramified" separable extensions of a local field $K$ and the category of "at most $e$-ramified" extensions of the length $e$ truncation of the ring of integers of $K$.
The main point of all this is that the behaviour of objects defined over discrete valuation rings is often determined by their reduction modulo a power of the maximal ideal i.e. on truncated data. It also ties in with Krasner's idea (hence the title of Deligne's paper) that local fields of characteristic $p$ are limits of local fields of characteristic 0 as the absolute ramification index tends to infinity.