I was reading the following lemma

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However I cannot understand what "Galois group of $F$" means. I usually look at a Galois group of an extension of fields.

Note: the previous result says that if $F$ is loclaly finite and quadratically closed and $[F:E]$, then $E$ is still quadraticlaly closed.

Reference: Sylow Theory, Formations and Fitting Classes in Locally Finite Groups. By (author): Martyn R Dixon

  • 2
    $\begingroup$ I think it means the Galois group of $F$ over its characteristic field $\Bbb F_p$. Although, the proof seems to be correct for the Galois group of $F/E$ for any subfield $E$ (for which the extension is Galois). $\endgroup$ – Greg Martin Nov 24 '16 at 7:38
  • $\begingroup$ @GregMartin I also thought that it could be $\mathbb{F}$ over $\mathbb{F}_p$, but the proof, I think, only works for finite Galois extension. Why you say that works also for an arbitrary Galois extension? Furthermore seems it to be working on even characteristic? $\endgroup$ – W4cc0 Nov 24 '16 at 8:53
  • $\begingroup$ Ok, I think by Artin's Lemma everything work fine. However, I think it is still not necessary to assume odd characteristic. What do you think? $\endgroup$ – W4cc0 Nov 24 '16 at 8:57
  • $\begingroup$ Perhaps the "previous result" requires it? $\endgroup$ – Greg Martin Nov 24 '16 at 18:55
  • $\begingroup$ Nope, the statement of the previous result is exactly as in the note I wrote in the question: in fact in the proof he distinguish the two cases also (even and odd). $\endgroup$ – W4cc0 Nov 24 '16 at 19:46

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