On page 595 of Dummit and Foote, there is an example of the field $\overline{\mathbb{F}}_p(x,y)$ of rational functions in the variables $x$ and $y$ over the algebraic closure $\overline{\mathbb{F}}_p$ of $\mathbb{F}_p$, and that this is not a simple extension of the subfield $F=\overline{\mathbb{F}}_p(x^p,y^p)$.
The authors state that "it is easy to see that $[\overline{\mathbb{F}}_p(x,y): \overline{\mathbb{F}}_p(x^p,y^p)]=p^2$. However, I am having trouble seeing why this is true. What would the basis for this extension be? Or is there an easier way to see why this is the degree?