Studying Galois i encountered the following problem. And i am really stuck on that one. What would be the best way to solve it?
Let $k$ be a field of characteristic $p>0$ and let $K=k(Y,Z)$ be the field of rational functions in two variables. Show that the polynomial $f=X^{p^2}+YX^p+Z\in K[X]$ is irreducible. Let $L=K(\alpha)$ with $\alpha$ a root of $f$. Prove that there exists no subfield $M$ of $L$ with $K\subset M\subset L$, $L/M$ separable, $M/K$ purely inseparable.
I am guessing i should consider the minimal polynomial of $\alpha$ over M, but i haven't managed to get anything useful
