Let $K$ be a field with characteristic $p>0$ and $M=K(X,Y)$ the field of rational functions in 2 variables over $K$. We consider the subfield $L=K({X}^{p},{Y}^{p})\subset M$.

Show that $[M:L]={p}^{2}$.

I guess I need the property that $[K(x):K({x}^{n})]=n$, which we showed already. But I actually do not know how to use it here. I can not work well with that field in 2 variables..

Show that $[M:L]={p}^{2}$.

I guess I need the property that $[K(x):K({x}^{n})]=n$, which we showed already. But I actually do not know how to use it here. I can not work well with that field in 2 variables..