Timeline for Fields such that every finite Galois extension is solvable
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16 events
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| Sep 15, 2021 at 16:45 | comment | added | Arno Fehm | @YCor: Yes, which such groups occur as absolute Galois groups and how the fields with these absolute Galois groups look like are two different questions. Note though that at least for the abelian case the work of Koenigsmann (on solvable absolute Galois groups) does give at least some information on these fields: Unless the group is projective, the field is either formally real or admits a nontrivial henselian valuation. | |
| Sep 15, 2021 at 11:34 | comment | added | YCor | OK. Actually, in the abelian case, the question which profinite abelian groups occur as absolute Galois groups is fully answered in this answer: namely the cyclic group of order 2 and all torsion-free ones (which are classified: possibly infinite products of $p$-adics groups for primes $p$ allowing arbitrary multiplicities). However this says little about the classification of fields with this property. | |
| Sep 14, 2021 at 21:16 | comment | added | Arno Fehm | As YCor says, the question could be rephrased as asking which prosolvable groups occur as absolute Galois groups. Koenigsmann (Solvable absolute Galois groups are metabelian, Invent. Math. 144, 2001) classified solvable absolute groups, but I don't think one has a complete answer to the prosolvable version. | |
| Sep 14, 2021 at 13:02 | comment | added | Buckeye | Sorry I meant that there should be at least one field but we don't care if there is a lot of different fields realizing the same profinite group | |
| Sep 14, 2021 at 13:00 | comment | added | YCor | If one takes a $p$-Sylow in the absolute Galois group of $\mathbf{Q}$, it seems it corresponds to an extension whose absolute Galois group is pro-$p$. These groups are not solvable, see doi.org/10.1016/j.aim.2015.05.017 | |
| Sep 14, 2021 at 12:55 | history | edited | YCor |
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| Sep 14, 2021 at 12:55 | comment | added | YCor | If you only consider the group, it's plainly a question about profinite groups and the answer is "prosolvable groups", by definition (which more properly should be called "pro-(finite solvable)"). This, in a sense, makes the question disappointing: the interest is when fields are of some interest. | |
| Sep 14, 2021 at 12:52 | comment | added | Buckeye | It is possible for different fields to have the same absolute Galois group, I think it emphasizes the groups more clearly | |
| Sep 14, 2021 at 12:50 | comment | added | YCor | Isn't the "more pointed" question just a restatement? (apart from excluding abelian cases). | |
| Sep 14, 2021 at 12:36 | history | edited | Buckeye | CC BY-SA 4.0 |
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| Sep 14, 2021 at 12:28 | history | edited | Buckeye | CC BY-SA 4.0 |
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| Sep 14, 2021 at 11:41 | comment | added | Wojowu | Related question for fields with abelian absolute Galois groups - no complete answer, but some hopefully helpful references. | |
| Sep 14, 2021 at 11:40 | comment | added | Wojowu | Separably closed fields, finite fields, quasi-finite fields,... I doubt there is any reasonable classification. | |
| Sep 14, 2021 at 11:40 | comment | added | Jason Starr | Finite fields. . . . . | |
| S Sep 14, 2021 at 11:35 | review | First questions | |||
| Sep 14, 2021 at 12:23 | |||||
| S Sep 14, 2021 at 11:35 | history | asked | Buckeye | CC BY-SA 4.0 |