Let $F$ be the field $\mathbb{F}_{p}(t,u)$ of rational functions in two variables $t$ and $u$ over the finite field $\mathbb{F}_{p}$. Let $L=F(T)$ be the field extension of $F$ generated by a root $T$ of $f(X)=X^{p}-t\in F[X]$, and let $M=L(U)$ be the field extension of $L$ generated by a root $U$ of $g(X)=X^{p}-u\in L[X]$. Prove that $M/F$ is an extension of degree $p^{2}$ and $M$ is the field $\mathbb{F}_{p}(T,U)$ of rational functions in two variables $T$ and $U$ over the finite field $\mathbb{F}_{p}$. Prove that $M$ cannot be generated by a single element over $F$.
This is a `homework' question (optional extra) and I am not looking for a worked solution, just a hint on how to go about starting it.
