Recently, I found a paper by Schilling http://www.jstor.org/pss/2371426, 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?

  • $\begingroup$ What do you mean by an "infinite field of algebraic numbers"? Class field theory, as usually understood, studies abelian extensions of algebraic number fields. $\endgroup$ – Vladimir Dotsenko Apr 2 '10 at 15:00
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    $\begingroup$ Ah, you mean infinite-dimensional extensions? The ambiguity your title creates is mildly annoying. $\endgroup$ – Vladimir Dotsenko Apr 2 '10 at 15:06
  • $\begingroup$ 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. $\endgroup$ – abcdxyz Apr 2 '10 at 15:07

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

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