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Is there an analogous theory to Galois extension of fields for commutative rings? In particular, what does it mean for a ring extension to be Galois? Thanks.

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There is an analogy for finite \'etale ring extensions (this is a special case of and the motivation for Grothendieck's Galois theory (see en.wikipedia.org/wiki/Grothendieck%27s_Galois_theory). Namely, just as if $k$ is a field, then there is an equivalence of categories between continuous transitive $G$-sets ($G$ the absolute Galois group) and algebraic extensions of $k$, there is an equivalence of categories between finite \'etale maps $R \to S$ ($R$ a fixed ring, say a domain) and finite continuous $G$-sets for $G$ the \'etale fundamental group. –  Akhil Mathew Jul 2 '11 at 18:15
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There is not one but many such theories. See e.g. mathoverflow.net/questions/63741/… –  Pete L. Clark Jul 2 '11 at 18:18
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3 Answers 3

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In several algebra books there are chapters devoted to integral ring extensions. (I refer to Lang's Algebra, Chapter VII "Extensions of Rings".) Here there are discribed "integrally closed" rings $A$ and there field of fractions $K$, together with a Galois extension $L$ over $K$ and the integral closure $B$ of $A$ in $L$. Instead of the roots of irreducible polynomials in $K$, that are split in $L$, the automorphisms of the Galois group now permute prime ideals of $B$ lying above a fixed prime ideal in $A$.

These kind of ring extensions have several more interesting properties. For details and the definitions of mentioned notions are given in the cited text.

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Thanks Peter, this seems like exactly what I was looking for. –  B M Jul 2 '11 at 18:23
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After reading the other answers, this is not necessarily a Galois theory for rings but rather the Galois Theory of fields applied to rings. Nevertheless, this topic is very interesting and at the foundation of algebraic number theory, so very well worth looking into. (Btw if this is the answer you were looking for, you can upvote it and give it the answer tick. ;) ) –  Peter Patzt Jul 2 '11 at 18:32
    
After reading through Lang, it seems, I am more interested in this. I attended a lecture which briefly mentioned this, so I formulated a vague question. –  B M Jul 3 '11 at 4:06
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See the following paper for an accessible introduction, and see also the answers to the MO question Is there a Galois correspondence for ring extensions?

M. Ferrero; A. Paques. Galois Theory of Commutative Rings Revisited.
Contributions to Algebra and Geometry, 38 (1997), No. 2, 399-410.

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Thanks, Bill. I will check out these two references. –  B M Jul 2 '11 at 18:24
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You should check Galois Theories of Francis Borceux.

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