The whole question looks like-
Let $F, E$ and $K$ be fields such that $F\subseteq E\subseteq K$ and $K$ is normal over $F$. Show that any $F$-homomorphism $\sigma :E\to K$ can be extended to an $F$-automorphism of $K$.
Now, first of all I have to extend a homomorphism $\sigma$ into an automorphism. So, here $\sigma$ must be injective. So, we have to first prove the injectiveness of $\sigma$. Am I right?
But I cannot extend this $F$-homomorphism to an $F$-automorphism. How to extend it?
Can anybody help me in this regard? Thanks for assistance in advance.