A field $F$ is separably closed if whenever $\alpha\in\bar{F}$ is separable over $F$ we have $\alpha\in F$. A separable closure of $F$ is a field $E\supset F$ such that $E$ is separably closed and $E/F$ is separable.
Now suppose that $E/F$ is normal and $K$ is the separable closure of $F$, how can I prove that $E/K$ is purely inseparable?
I was trying to prove that if we take $\alpha\in E$ and an homomorphism $\tau: K(\alpha)\rightarrow \bar{K}$ that fixes $K$ then it is the identity, but I got stuck, any help?
Another thing that is not so clear is how do we know that $K\subset E$?