Suppose $L/K$ is not separable. Does this imply that there is an $a\in L$ such that $a^p\in K$ but $a\not\in K$?
I know that the characteristic of the fields must necessarily be $p>0$. By definition, there must be some $a\in L$ that is inseparable; that is, it must have a minimal polynomial that has multiple roots in a field extension. Equivalently, its derivative must be $0$. Where to go from here?
