# Let K be a field with characteristic p &gt; 0 and M = K ( X ,

Let $K$ be a field with characteristic $p>0$ and $M=K\left(X,Y\right)$ the field of rational functions in 2 variables over $K$. We consider the subfield $L=K\left({X}^{p},{Y}^{p}\right)\subset M$.
Show that $\left[M:L\right]={p}^{2}$.
I guess I need the property that $\left[K\left(x\right):K\left({x}^{n}\right)\right]=n$, 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..
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lywiau63
Recall that for some field $J$ so that $L\subset J\subset M$ you have that the degree of the extension $L\subset M$ is the product of the degrees of the extensions $L\subset J$ and $J\subset M$.
Use this for example with $J=K\left({X}^{p},Y\right)$, applying the result you know twice.