Separables elements of extension

Question

Pleas help me to Find the separable element of the extension $F_p(X,Y)/F_p(X^p,Y^p)$. Thanks.

Answer 1

If $a=\frac{u(X,Y)}{v(X,Y)} \in \Bbb{F}_p(X,Y)$ (where $ u,v \in \Bbb{F}_p[X,Y]$) then $a^p =\frac{u(X^p,Y^p)}{v(X^p,Y^p)}\in \Bbb{F}_p(X^p,Y^p)$

$a$ is a root of $T^p - a^p = (T-a)^p$ having only one root.

Whence $\Bbb{F}_p(X,Y)/\Bbb{F}_p(X^p,Y^p)$ is purely inseparable.

$\Bbb{F}_p(X^p)(Y)/\Bbb{F}_p(X^p)(Y^p)$ is purely inseparable of degree $p$ and $\Bbb{F}_p(Y)(X)/\Bbb{F}_p(Y)(X^p)$ is purely inseparable of degree $p$ thus $\Bbb{F}_p(X,Y)/\Bbb{F}_p(X^p,Y^p)$ is purely inseparable of degree $p^2$