$F/K$ finite extension, then $\exists !$ intermediate field $K \subset L \subset F$ such that $L/K$ separable and $F/L$ purely inseparable

Suppose that $L/K$ is a finite separable extension, and $E/K$ is an arbitrary extension. Then $\mathrm{Hom}_K(L,E)$ consists of at most $[L:K]$ elements, because a field homomorphisms $\sigma :L\rightarrow E$ that fixes $K$ is uniquely determined by the image $\sigma (x)$ of a primitive element of $L/K$, which in turn must be a root of the minimal polynomial $f$ of $x$ over $K$.

To characterize separability you could say that the finite extension $L/K$ is separable if and only if there exists a field extension $E/K$ such that $|\mathrm{Hom}_K(L,E)|=[L:K]$. (For $E$ you can then take the normal hull of $L/K$, or any normal extension $N/K$ containing $L$.)