I want to show for $K$ a field of characteristic $p$, that $[K(X):K(X^p-X)]=p$.
Say $X^p-X=Y$. I want to argue something along the lines of $Z^p-Y-a \in K(Y)[Z]$ is irreducible, if $a$ is a root, then it has a minimal polynomial of degree $p$, so $[K(Y)(a):K(Y)]=p$. Then $X=a$ works, so $K(Y)(a)=K(X)$, giving the result.
But I’m not quite sure how to make this argument clear or if it works. What can I do?