I got this exercise: Let $p>2$ be a prime number, and let $F$ be a finite field of characteristic $p$. Find the largest field $E$ in between $F(X)$ and $F(X^{1/2p})$ such that $E$ is a separable extension over $F(X)$.
It seems that I need to find all elements in $F(X^{1/2p})$ that are separable over $F(X)$. I know that a finite field of characteristic $p$ is isomorphic to $\mathbf{F}_{p^n}$, but I am not sure how this would help solving the problem. Any help would be appreciated.