Dayanara Terry

2022-06-29

Let $K$ be a field with characteristic $p0$ 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 $[M:L]={p}^{2}$.

We 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..

lywiau63

Beginner2022-06-30Added 13 answers

Keep in mind that in some fields $J$ so that $L\subset J\subset M$ You are in possession of the extension's degree $L\subset M$ is calculated as the product of the extensions' degrees. $L\subset J$ and $J\subset M$.

Use this for example with $J=K({X}^{p},Y)$, applying the outcome you are aware of twice.

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