I've seen the following claim in my notes, but I couldn't see why it's true: Suppose that $y \in F_p((x))$ is transcendent over $F_p(x)$, denote $L:=F_p(x, y)$ and let $L^p$ be the field of $p$th powers of $L$.
My question is:
Why is $[L : L^p]=p^2$?