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.