Let $G$ be a finite group of automorphisms of $E$ and set $F=Fix(G)$, Then why is $E:F$ always separable ?
I have a feeling that it has something to do with the idea that if $E:F$ is separable hence all elements adjoined in $E$ are separable and so their min. poly has simple roots, and so if $E:F$ were not separable it would mean that $\alpha\in E$ appears twice in its min poly and so the co-efficents of the min poly would contain some power of $\alpha$ and then would not be fixed by G.
I'm not sure if I'm thinking in the right vein at all though , I'm finding it difficult to formulate the reason concisely in my head .
Could anyone clear this up for me please ?