Talk:Galois group

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[edit] Typo?

Hello,

I'm far from being certain of what I'm going to say but isn't there a problem when saying: "automorphisms of E/F - that is, isomorphisms α from E to itself"

Shouldn't it be "automorphisms of E - that is, isomorphisms α from E to itself" or "automorphisms of E/F - that is, isomorphisms α from E/F to itself"

Randomblue 22:34, 24 July 2007 (UTC)

The punctuation was wrong (I've fixed it now). By an automorphism of E/F we mean an automorphism of E that fixes F pointwise. --Zundark 07:24, 25 July 2007 (UTC)

[edit] Gal and Aut

I note that the author has been careful to use Aut instead of Gal in the case of R/Q. However, are we sure that Gal is appropriate in the line below ("Gal(C/Q)"). I am only knowledgeable on Galois theory of finite extensions and may be overlooking some aspect of the theory that develops infinite extensions, but C is not the splitting field for any polynomial of any degree (even countably infinite) over Q. Is it possible that Gal here is a typo and should be Aut as in the case of R? —Preceding unsigned comment added by Gtg207u (talk • contribs) 05:42, 7 December 2007 (UTC)

I think C/Q is a Galois extension according to the definition given in that article. --Zundark (talk) 15:41, 7 December 2007 (UTC)
It fails even the basic precondition of being an algebraic extension. -- EJ (talk) 16:46, 7 December 2007 (UTC)
True - I hadn't noticed that someone (you) had added that restriction to the first paragraph. The condition is still not present in the definition itself - perhaps you should add it there too. --Zundark (talk) 17:55, 7 December 2007 (UTC)
Yes, I added the restriction. I thought it was simply omitted: the definition without the restriction was at odds with the standard usage of the term (to the best of my knowledge), as well as its usage in other WP articles (e.g., profinite group, glossary of field theory, or this very article). In particular, the fundamental theorem of Galois theory breaks badly under the unrestricted definition, which makes the definition kind of pointless (IMHO), and directly contradicts statements to the contrary in several WP articles.
I've reformulated the definition now to make it more explicit. -- EJ (talk) 15:07, 10 December 2007 (UTC)
"It can be shown that E is algebraic over F if and only if the Galois group is pro-finite." is a nice fact from this article, so one might want to decide what a galois extension is like when there are transcendental elements. Obviously we cannot replace galois group with field automorphism group, so something in the fact depends on the extension being galois. Perhaps K=k[x] is galois over k, and F is galois over k if F is galois over K and K is galois over k?
At any rate, the Aut/Gal confusion is widespread, so be careful in what might seem like minor changes. JackSchmidt (talk) 18:00, 7 December 2007 (UTC)
The statement you mention is certainly puzzling. As it is written, it actually requires every field extension to have a Galois group. That's probably unintended, but even in the reading "a Galois extension is algebraic iff bla bla" it's unclear what is meant by a Galois extension. I suspect it's actually possible the statement is in error, so at any rate I'd like to see a clear reference for that. -- EJ (talk) 15:07, 10 December 2007 (UTC)
I checked the page history, and the fact was in the original version of the page. I think for any field k, Aut(k(x)/k)=PGL(2,k). For k=Z/2Z it is finite and so pro-finite. If there are *any* non-algebraic extensions that are allowed to be "galois", then I would think k(x)/k would be. For infinite fields, PGL(2,k) is infinite and rarely pro-finite, so perhaps that is what was meant. Making such a statement (true and) precise would be fairly difficult, I think. JackSchmidt (talk) 17:34, 10 December 2007 (UTC)
Oh and to be clear, I am not trying to say the definition of Galois extension needs to achieve the maximum level of abstruse generality to apply equally and meaninglessly to all cases, just that when changing (or clarifying) its generality to be careful of all the places it might break. Honestly, I think the fact from Galois group should have just indicated more clearly what sort of context it meant, and that Galois extension should primarily apply to finite dimensional extensions, with infinite dimensional algebraic extensions as a generalization, and whatever the fact intends as yet another example of how to extend the definition. JackSchmidt (talk) 18:18, 7 December 2007 (UTC)
I agree, though it's worth pointing out that the generalization to infinite algebraic extensions works literally the same way as for finite extensions in many cases (including the definition itself), so one can as well state such stuff directly for all algebraic extensions, without separating the finite-dimensional case first. -- EJ (talk) 15:07, 10 December 2007 (UTC)

[edit] Beneficial to mention a certain theorem?

Does anyone think it might benefit to mention (I believe it's called) the Conjugation Isomorphism Theorem? It basically says that if two numbers a and b algebraic over a field, then a mapping defined between the field extensions obtained by ajoining a and b which sends Σiciai to Σicibi is an isomorphism if and only if a and b are conjugate over the original field. This theorem has some useful corollaries.LkNsngth (talk) 05:11, 6 April 2008 (UTC)