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# Is there a notion of Galois extension for $\mathbb{Z}Z / p^2$?

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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 $\mathbb{Z}_p$)?

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