Every normal extension has a subfield which is a galois extension?

Question

Let $F:E$ be a normal and finite field extension. I was asked to show that there is an intermediate field $E \subset K \subset F$ such that $K:E$ is a gextension, but I'm skeptical if this result is even valid: If everything here was separable, then there is nothing to show, the extension $F:E$ is galois, but if it's not, how can I come up with a separable smaller field that would make the extension galois?