Talk:Fundamental theorem of Galois theory

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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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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

Improve Article Request

Can you improve this article? Well doing please. Nassiv64 (talk) 20:31, 6 March 2025 (UTC)[reply]

Every article can be improved. Which changes do you have in mind? For such a request you must state clearly which changes you are asking for. D.Lazard (talk) 11:03, 7 March 2025 (UTC)[reply]

Yes, do add Example 5. Nassiv64 (talk) 12:23, 7 March 2025 (UTC)[reply]

Which example 5? I guess that you are asking to add your homework to Wikipedia for avoiding to work yourself. D.Lazard (talk) 12:27, 7 March 2025 (UTC)[reply]

I don’t have ideas for adding example 5 section. Nassiv64 (talk) 13:21, 7 March 2025 (UTC)[reply]

Four examples seem sufficient. D.Lazard (talk) 15:29, 7 March 2025 (UTC)[reply]

okay. Nassiv64 (talk) 15:52, 7 March 2025 (UTC)[reply]