10
$\begingroup$

I am working on this exercise:

If $E$ is an intermediate field of an extension $F/K$ of fields. Suppose $F/E$ and $E/K$ are Galois extensions, and every $\sigma\in Gal(E/K)$ is extendible to an automorphism of $F$, then show that $F/K$ is Galois.

I can see that any $\sigma$ extended over $F$ fixes elements in $K$ but not in $E-K$. But how to show it doesn't fix elements in $F-E$?

Hints only please, this is homework.

p.s. we use Kaplansky, he doesn't require Galois extensions to be finite dimensional.

  • 0
    Is your definition of Galois that the fixed field of $\operatorname{Aut}(L/k)$ is $k$? If so, then it seems helpful to note that $\operatorname{Gal}(F/E) \subset \operatorname{Aut}(F/K)$.2012-02-10
  • 0
    i solved it. You can remoe this question.2012-02-10
  • 0
    You have the power to remove it, I think, but I don't think there's any harm in leaving it up.2012-02-10
  • 14
    @Allan: Instead of removing the question, post your solution as an answer!2012-02-10

2 Answers 2