Talk:Fundamental theorem of Galois theory

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 $f(\theta )=\omega \cdot \theta$, where $\theta$ is a cube root of 2 and $\omega$ is a primitive cube root of unity. Plclark (talk) 00:36, 9 February 2008 (UTC)

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)

Example 3 and geometric cross-ratio

The elements of the group described in example 3 take the same form as geometric cross-ratios. I think they should be cross referenced somehow. — Anita5192 (talk) 22:03, 8 May 2015 (UTC)

At the moment it is referenced indirectly; if you click on the link J-invariant#Alternate Expressions you'll immediately see the same set G and the reference to cross-ratio. MvH (talk) 13:01, 9 May 2015 (UTC)

I added a direct link to six cross-ratios. The link says G isom S3, so the the argument for G isom S3 (the action on 0,1,infinity) is not needed anymore. MvH (talk) 13:11, 9 May 2015 (UTC)

Thank you! I think the connection is clearer now. — Anita5192 (talk) 04:05, 10 May 2015 (UTC)