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Recently, I found a paper by Schilling, which mentions that for certain infinite field of algebraic numbers there is an analog of class field theory. By infinite field of algebraic number we mean an infinite extension of $\mathbb{Q}$. The paper cite a previous paper by Moriya which was the origin of the idea. I could not read the later since it is in German. Since the first paper is quite old (1937), I believe there must have been a lot of development in the mean time.

My question: Do we have an analog of class field theory over an arbitrary infinite field of algebraic number?

An even more general question: Do we have an analog of class field theory over an arbitrary field. This seems a bit greedy, but since we know that an algebraic closed field of characteristic 0 is totally characterized by its trancendence degree so if the answer to the previous question is positive the answer to this is perhaps not too far. Am I making sense?

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What do you mean by an "infinite field of algebraic numbers"? Class field theory, as usually understood, studies abelian extensions of algebraic number fields. – Vladimir Dotsenko Apr 2 '10 at 15:00
Ah, you mean infinite-dimensional extensions? The ambiguity your title creates is mildly annoying. – Vladimir Dotsenko Apr 2 '10 at 15:06
Sorry, it was the term used by the author of the paper so I just follow it. I thought it was a standard name. I will update asap. – abcdxyz Apr 2 '10 at 15:07
up vote 8 down vote accepted

Iwasawa theory studies abelian extensions of fields $K$ where $K$ is a $\mathbb{Z}_p$-extension of $\mathbb{Q}$, that is the Galois group of $K/\mathbb{Q}$ is $\mathbb{Z}_p$. The corresponding Galois groups (of extensions of $K$) and class groups (of really subfields) of $K$, suitably interpreted, become $\mathbb{Z}_p$-modules and there are interesting relations between these modules and $p$-adic $L$-functions. It is a vast subject. Washington's book, Introduction to Cyclotomic Fields, is a good entry point.

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Suitably interpreted, they are even $\mathbb{Z}_p [\mathbb{Z}_p]$-modules. – Dror Speiser Apr 2 '10 at 17:25
And even ${\bf Z}_p[[{\bf Z}_p]]$-modules. – Chandan Singh Dalawat Oct 7 '10 at 13:55

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