# Galois over Galois

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.

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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 $\operatorname{Gal}(F/E) \subset \operatorname{Aut}(F/K)$. –  Dylan Moreland Feb 10 '12 at 14:19
Thank you Dylan. –  Allan Feb 10 '12 at 15:22
i solved it. You can remoe this question. –  Allan Feb 10 '12 at 15:27
You have the power to remove it, I think, but I don't think there's any harm in leaving it up. –  Dylan Moreland Feb 10 '12 at 15:30
@Allan: Instead of removing the question, post your solution as an answer! –  Arturo Magidin Feb 10 '12 at 16:56