Is there a Galois correspondence for ring extensions? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T20:15:15Z http://mathoverflow.net/feeds/question/63741 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63741/is-there-a-galois-correspondence-for-ring-extensions Is there a Galois correspondence for ring extensions? Mauricio 2011-05-02T21:19:29Z 2011-06-27T02:17:14Z <p>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.</p> <p>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?</p> <p>Thanks in advance!</p> http://mathoverflow.net/questions/63741/is-there-a-galois-correspondence-for-ring-extensions/63742#63742 Answer by Mariano Suárez-Alvarez for Is there a Galois correspondence for ring extensions? Mariano Suárez-Alvarez 2011-05-02T21:22:44Z 2011-05-02T21:39:54Z <p>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. <a href="http://www.ams.org/mathscinet-getitem?mr=MR0195922" rel="nofollow">MR0195922</a> (33 #4118)] The theory developed there does include a Galois correspondence.</p> <p>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.</p> http://mathoverflow.net/questions/63741/is-there-a-galois-correspondence-for-ring-extensions/63818#63818 Answer by Anatoly Kochubei for Is there a Galois correspondence for ring extensions? Anatoly Kochubei 2011-05-03T15:20:56Z 2011-05-03T15:20:56Z <p>In addition to the above references, I would like to mention some non-commutative extensions of the Galois theory. See </p> <p>P. M. Cohn, Skew Fields, Cambridge University Press, 1995</p> <p>for the Galois theory of skew fields. Extensions to some classes of noncommutative rings are given in the book</p> <p>V. K. Kharchenko, Noncommutative Galois theory, Novosibirsk, 1996,</p> <p>available only in Russian, and many papers of its author, some of which exist also in the English translation.</p> http://mathoverflow.net/questions/63741/is-there-a-galois-correspondence-for-ring-extensions/66287#66287 Answer by Jizhan Hong for Is there a Galois correspondence for ring extensions? Jizhan Hong 2011-05-28T15:53:30Z 2011-06-27T02:17:14Z <p>For a "survey" of Galois theory of commutative rings, there is one book:</p> <p><em>The Separable Galois Theory Commutative Rings</em> by Andy R. Magid (1974). </p> <p>which has a nice section summarizing the state of the development up to 1974. </p> <p>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:</p> <p><em>Galois Theories</em> by Francis Borceux and George Janelidze (2001). </p> http://mathoverflow.net/questions/63741/is-there-a-galois-correspondence-for-ring-extensions/66328#66328 Answer by SGP for Is there a Galois correspondence for ring extensions? SGP 2011-05-29T02:23:31Z 2011-05-29T02:23:31Z <p>see <a href="http://mathoverflow.net/questions/31623/an-advanced-exposition-of-galois-theory" rel="nofollow">related question</a></p> <p>and <a href="http://arxiv.org/abs/math/0206203" rel="nofollow">SGA1</a> as well as <a href="http://websites.math.leidenuniv.nl/algebra/GSchemes.pdf" rel="nofollow">Lenstra's notes</a> </p> <p>and <a href="http://arxiv.org/abs/1006.2562" rel="nofollow">Manjul Bhargava and Matt Satriano's paper</a> </p>