I was reading the following lemma
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