Is there a Galois correspondence for ring extensions? - MathOverflow

Is there a Galois correspondence for ring extensions?

2011-05-02T21:19:29Z

Given an ring extension of a (commutative with unit) ring, Is it possible to give a "good" notion of "degree of the extension"?. By "good", I am thinking in a degree which allow us, for instance, to define finite ring extensions and generalize in some way the Galois' correspondence between field extensions and subgroups of Galois' group.

I suppose one can call a ring extension $A\subset B\ $ finite if $B$ is finitely generated as an $A$-module, and the degree would be the minimal number of generators, but is that notion enough to state a correspondence theorem?

Thanks in advance!

Answer by Mariano Suárez-Alvarez for Is there a Galois correspondence for ring extensions?

2011-05-02T21:22:44Z

There is indeed a theory of Galois extension of rings. See, for example, the very nice paper [Chase, S. U.; Harrison, D. K.; Rosenberg, Alex. Galois theory and Galois cohomology of commutative rings. Mem. Amer. Math. Soc. No. 52 1965 15--33. MR0195922 (33 #4118)] The theory developed there does include a Galois correspondence.

There is even a Hopf-Galois theory, where the Galois group is replaced by a Hopf algebra (co)acting on the big ring, for extra fun---the correspondence in this case, though, is quite more delicate/complicated.

Answer by Anatoly Kochubei for Is there a Galois correspondence for ring extensions?

2011-05-03T15:20:56Z

In addition to the above references, I would like to mention some non-commutative extensions of the Galois theory. See

P. M. Cohn, Skew Fields, Cambridge University Press, 1995

for the Galois theory of skew fields. Extensions to some classes of noncommutative rings are given in the book

V. K. Kharchenko, Noncommutative Galois theory, Novosibirsk, 1996,

available only in Russian, and many papers of its author, some of which exist also in the English translation.

Answer by Jizhan Hong for Is there a Galois correspondence for ring extensions?

2011-05-28T15:53:30Z

For a "survey" of Galois theory of commutative rings, there is one book:

The Separable Galois Theory Commutative Rings by Andy R. Magid (1974).

which has a nice section summarizing the state of the development up to 1974.

There is also a more general book aiming at a topos-theory style general Galois theory (although I haven't read it) including also a nice survey:

Galois Theories by Francis Borceux and George Janelidze (2001).

Answer by SGP for Is there a Galois correspondence for ring extensions?

2011-05-29T02:23:31Z

see related question

and SGA1 as well as Lenstra's notes

and Manjul Bhargava and Matt Satriano's paper