Let $p$ a prime ad $\mathbb{F}_p$ the finite field with $p$ Elements. In want to understand why the field extension $\mathbb{F}_p(X,Y) \subset \mathbb{F}_p(X^{1/p},Y^{1/p})$ isn't a simple extension. In other words why there no $a \in \mathbb{F}_p(X^{1/p},Y^{1/p})$ with $\mathbb{F}_p(X^{1/p},Y^{1/p}) = \mathbb{F}_p(X,Y)(a)$.
Any ideas?
Motivation: that seems to be an example for a finite normal, non separable extension which isn't simple.