Timeline for Automorphism groups of the complex numbers, and other fields
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| S Mar 11, 2021 at 13:07 | history | bounty ended | CommunityBot | ||
| S Mar 11, 2021 at 13:07 | history | notice removed | CommunityBot | ||
| S Mar 3, 2021 at 11:10 | history | bounty started | THC | ||
| S Mar 3, 2021 at 11:10 | history | notice added | THC | Canonical answer required | |
| S Nov 7, 2020 at 0:06 | history | bounty ended | CommunityBot | ||
| S Nov 7, 2020 at 0:06 | history | notice removed | CommunityBot | ||
| Oct 30, 2020 at 5:45 | comment | added | Emil Jeřábek | @LSpice Indeed, they need not. | |
| Oct 29, 2020 at 23:18 | comment | added | LSpice | @EmilJeřábek, so automorphisms of $\overline{\mathbb Q}$ need not extend to $\mathbb C$ without choice? (I know that the standard proofs of extension use choice, but I didn't know if it was essential.) | |
| S Oct 29, 2020 at 22:21 | history | bounty started | THC | ||
| S Oct 29, 2020 at 22:21 | history | notice added | THC | Canonical answer required | |
| Oct 29, 2020 at 22:21 | history | edited | THC | CC BY-SA 4.0 |
added 252 characters in body
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| Oct 29, 2020 at 15:46 | comment | added | მამუკა ჯიბლაძე | @EmilJeřábek Did you have in mind this question back from 2010? It currently is on the "Related" list here. | |
| Oct 23, 2020 at 14:49 | comment | added | Emil Jeřábek | @მამუკაჯიბლაძე I seem to vaguely recall that there was a question whether it is consistent with ZF that Aut($\mathbb C$) has size strictly between $2$ and $2^{2^\omega}$, with inconclusive answers (but I can’t find anything). I believe that at least the Artin–Schreier theorem can be made choiceless enough to show that if there are more than 2 automorphisms, there are infinitely many. However, none of this should concern $\mathrm{Gal}(\tilde{\mathbb Q}/\mathbb Q)$, which should be as wild as usual even in ZF, as $\tilde{\mathbb Q}$ is a countable (and therefore well ordered) field. | |
| Oct 23, 2020 at 13:11 | comment | added | მამუკა ჯიბლაძე | @AsafKaragila Thanks for the explanation. So the correct statement is something like "In some model of ZF (without C) there are exactly two automorphisms of $\mathbb C$"? Can there be intermediate cases? Like models where the Galois group of $\mathbb Q$ is abelian, things like that? | |
| Oct 23, 2020 at 13:04 | comment | added | Asaf Karagila♦ | @მამუკაჯიბლაძე It is consistent with ZF that every automorphism of a Polish group is continuous, in that case only the identity and conjugations are automorphisms of $\Bbb C$. | |
| Oct 23, 2020 at 13:03 | comment | added | Asaf Karagila♦ | Not accepting AC is not nearly enough. You're saying "Here is a finite set; I do not accept the claim it is empty. Therefore, it has exactly 4 elements". | |
| Oct 23, 2020 at 12:34 | comment | added | მამუკა ჯიბლაძე | How are you going to prove that $\mathbb C$ has no more than two automorphisms? | |
| Oct 23, 2020 at 12:13 | history | asked | THC | CC BY-SA 4.0 |