Timeline for transcendental Galois theory
Current License: CC BY-SA 2.5
15 events
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| Jun 22, 2022 at 8:13 | history | edited | CommunityBot |
replaced http://math.uga.edu/~pete with http://alpha.math.uga.edu/~pete
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| Aug 2, 2011 at 5:57 | answer | added | Pete L. Clark | timeline score: 15 | |
| Aug 2, 2011 at 5:29 | comment | added | Pete L. Clark | @Sylvain: so I checked, and indeed there it is on p. 173 of the English copy of Bourbaki's "Algebra II". By the way, I think that what I do in my note does lead to a solution of this exercise: the definition of Dedekind allows us to replace $K$ with any larger subextension, so we may assume that $L = K(t)$ and then if $E/L$ were finite then $E/K$ would be the function field of an algebraic curve: I do that case! | |
| Aug 2, 2011 at 5:01 | comment | added | Pete L. Clark | @Sylvain: that's interesting, thanks. I'll take a look at it. | |
| Jun 10, 2011 at 20:55 | comment | added | Sylvain Bonnot | That's a very intriguing question... An exercise in Bourbaki Algebra (chapter 5 I believe) calls "Dedekind extensions" your Galois extensions. Unfortunately, they don't propose much in the interesting direction, except the following: "If E is a Dedekind extension of K and L a pure transcendental subextension of E, distinct from K and such that E is an algebraic extension of L, show that E has infinite degree over L." I tried to use this after reading your very interesting notes, but without success... | |
| Feb 18, 2011 at 4:53 | comment | added | awllower | @Anweshi: May I ask for an example such that the linear independence of characters does not occur? Thanks. | |
| Jan 30, 2011 at 21:57 | answer | added | M.G. | timeline score: 4 | |
| Jan 6, 2011 at 21:36 | comment | added | user10849 | I would argue that once one moves beyond etale extensions that one should not look at discrete Galois groups. E.g. Jacobson's theory for exponent purely inseparable extensions uses Lie algebras. For a Galois theory for extensions $E/F$ I would be more inclined to look at subalgebras of the derivations (or of the differential operators) of $E$ over $F$. See e.g. "On Galois correspondence between intermediate fields and closed derivation subalgebras" Teppei Kikuchi J. Math. Kyoto Univ. 23, (1983), 281-287. | |
| Jul 28, 2010 at 11:33 | comment | added | David Corwin | You might be interested in Chapter 8 of jmilne.org/math/CourseNotes/FT.pdf. He also references a book by Shimura. | |
| May 1, 2010 at 0:47 | comment | added | Pete L. Clark | @DS: No, it's not in Borceux and Janelidze. | |
| May 1, 2010 at 0:08 | comment | added | David Steinberg | I always though this kind of souped-up Galois theory was covered thoroughly in <a href="books.google.com/… Theories</a> by Borceaux & Janelidze. I never read the book though, so I am probably mistaken. | |
| Feb 20, 2010 at 7:06 | comment | added | Pete L. Clark | I would argue that curiosity is one of the deepest reasons to be interested in something! But if you are asking whether there is some other conjecture up my sleeve that would follow from this: no, it is not related to any of my "real" work. | |
| Feb 19, 2010 at 18:38 | answer | added | none | timeline score: 3 | |
| Jan 16, 2010 at 14:10 | comment | added | Anweshi | I suppose for transcendental extensions Artin's theorem on linear independence of characters will break down, and without this powerful theorem it will be hard to draw any consequences. Are there deeper reasons for you getting interested in transcendental Galois extensions, in addition to it being a curiosity? | |
| Jan 16, 2010 at 4:40 | history | asked | Pete L. Clark | CC BY-SA 2.5 |