Talk:Fundamental theorem of Galois theory

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hi there Charles Matthews, I don't know exactly how to contact you directly. As you can see I'm making some major changes to all the Galois theory pages. I am planning to put up a summary plus ideas for future work in the next few hours, probably on the Galois theory discussion page. --Dmharvey 13:40, 27 May 2005 (UTC)

You can leave messages at User talk:Charles Matthews. Charles Matthews 13:45, 27 May 2005 (UTC)

missing logic, or my problem?

I may be missing something, but in the discussion of the cube roots of 2, it was stated that f(x) = w *x is an automorphism. I was under the impression that the automorphism had to map rationals to rationals, so that w = w * 1 = f(1) = 1. How, then could be w be defined as one of the complex roots? Could you clarify this part of the writeup? (or am I being stupid?) CharlesTheBold 01:32, 31 July 2007 (UTC)

It doesn't say that f(x) = w*x for all x; it says that , where is a cube root of 2 and is a primitive cube root of unity. Plclark (talk) 00:36, 9 February 2008 (UTC)Plclark

Missing detail in example section

In the example section it says: "Each such automorphism must send √2 to either √2 or −√2, and must send √3 to either √3 or −√3". Why is this? It seems like sending √2 to √3 and vice versa (simultaneously) would still fix a. Maybe we could add a sentence or link explaining this statement? Luqui (talk) 07:21, 24 September 2010 (UTC)

The automorphisms in G can only permute the roots of any polynomial. (See Lemma 18.3 in Algebra, A Graduate Course by I. Martin Isaacs.) So G permutes {√2, -√2} (the roots of the polynomial X2 - 2) and {√3, -√3} (the roots of the polynomial X2 - 3). I added some explanatory text. Bender2k14 (talk) 05:21, 9 March 2011 (UTC)