I've completed an introductory course in Galois Theory, but feel my understanding of separability is poor. I think my confusions boil down to the following question:
What is the relationship between a separable extension $L/K$ and the space $\mathrm{Hom}_K(L,E)$ for some field $E$ in which $K$ can be embedded?
Any explanations or references would be greatly appreciated.
Thanks
EDIT: Perhaps this is a bad questions, since it is too vague. I'm trying to solve the following question:
Let $F/K$ be a finite extension. Show that there is a unique intermediate field $K \subset L \subset F$ such that $L/K$ is separable and $F/L$ is purely inseparable (i.e. $|\mathrm{Hom}_L(F,E)| \leq 1$ for every extension $E$ of $L$).
Firstly, if $\mathrm{char}K = 0$ then we must have $L = F$ and the conditions are satisfied. So now suppose $\mathrm{char}K = p > 0$.
I took a guess and said $L$ is probably the smallest field containing all the elements of $F$ which are separable over $K$. Then if $\alpha \in F$ with minimal polynomial $f_L$ over $L$, and $\theta: F \to E$ is a homomorphism that fixes $L$, then $\alpha$ must be mapped to a root of $\theta(f_L)$. So if each element of $F\backslash L$ has a minimal polynomial over $L$ with only one distinct root, then I can see that the $|\mathrm{Hom}_L(F,E)| \leq 1$ will be satisfied. We also know that $\alpha$ has minimal polynomial $f_K$ over $K$ which is not separable, and so we have $f_L | f_K$ and $f_K \in K[X^p]$.
I'm stuck where to go from here, though. Is my approach correct?