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What are the fields such that every finite Galois extension is solvable?

We have algebraically closed fields, real closed fields, p-adic fields. Anything else?

A more pointed question after comments:

Which non-abelian absolute Galois groups have only solvable finite quotients?

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    $\begingroup$ Finite fields. . . . . $\endgroup$
    – Jason Starr
    Commented Sep 14, 2021 at 11:40
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    $\begingroup$ Separably closed fields, finite fields, quasi-finite fields,... I doubt there is any reasonable classification. $\endgroup$
    – Wojowu
    Commented Sep 14, 2021 at 11:40
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    $\begingroup$ Related question for fields with abelian absolute Galois groups - no complete answer, but some hopefully helpful references. $\endgroup$
    – Wojowu
    Commented Sep 14, 2021 at 11:41
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    $\begingroup$ Isn't the "more pointed" question just a restatement? (apart from excluding abelian cases). $\endgroup$
    – YCor
    Commented Sep 14, 2021 at 12:50
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    $\begingroup$ 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. $\endgroup$
    – YCor
    Commented Sep 14, 2021 at 12:55

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