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Consider the absolute Galois group $G_{\mathbb{Q}} := \mathrm{Aut}(\overline{\mathbb{Q}}/\mathbb{Q})$.

As I understand it, the only torsion elements have order $2$ (by Artin-Schreier), and they are all conjugate.

My questions concern the structure of this conjugacy class.

  • What is the subgroup generated by all involutions ?

  • How do the (largest) elementary abelian $2$-subgroups of $G_{\mathbb{Q}}$ look like ?

If we do not accept the Axiom of Choice, does the Artin-Schreier result still apply, as well as the answers on the aforementioned questions ? (In this case, I define $\overline{\mathbb{Q}}$ as the usual countable algebraic closure of $\mathbb{Q}$ inside $\mathbb{C}$.)

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    $\begingroup$ The subgroup generated by all involutions is the Galois group of $\overline{\mathbb Q}$ over the maximal totally real extension of $\mathbb Q$ as this is the fixed field of all the complex conjugations. $\endgroup$
    – Will Sawin
    Commented Apr 20, 2023 at 12:58
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    $\begingroup$ Will Sawin answers the first question, and the structure of that subgroup is known (a result by Fried-Haran-Völklein). The abelian subgroups are also known, in particular every elementary 2-abelian subgroup is in fact cyclic (this I think is an older result by Geyer), see this answer: mathoverflow.net/a/352952/50351 $\endgroup$
    – Arno Fehm
    Commented Apr 20, 2023 at 13:38
  • $\begingroup$ @ArnoFehm: so they are finite of size 2 ? $\endgroup$
    – THC
    Commented Apr 20, 2023 at 13:45
  • $\begingroup$ @ArnoFehm: does Geyer use Choice to obtain that result ? $\endgroup$
    – THC
    Commented Jul 5, 2023 at 11:19

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