Let $L=\bar{\mathbb{F}}_{p}(x,y)$ (for $\bar{\mathbb{F}_{p}}$ an algebraic closure of the finite field) and let $L^{(p)}$ be the image of $L$ under the Frobenius map $L \rightarrow L $, where $g(x) \mapsto g(x)^{p}$. Show that $[L:L^{(p)}]=p^{2}$ and moreover that there are infinitely many distinct subfields $L^{(p)} \subset E \subset L$.
I have shown the first statement (can take as a basis $\{1,x,y,,xy,x^{2},y^{2},\dots,x^{p-1}y^{p-1},x^{p},y^{p}\}$ which will have size $p^{2}$, so the extension is of degree $p^{2}$).
I'm stumped for the second part though. What would be a good observation to get started? (I saw a similar example for $p=2$, not sure on how to generalize this though).
Thanks for the help!