If $F=K(u,v)$ with $u^p$,$v^p\in K$ and $[F:K]=p^2$, $\operatorname{char} K=p>0$, then $F$ is not a simple extension of $K$.
Greetings I'm trying to show this exercise from Hungerford's Algebra Chapter five section 6 exercise 15; but I'm stuck. the exercise says the following: Let $\operatorname{char} K=p>0$ and assume $F=K(u,v)$ where $u^p,v^p\in K$ and $[F:K]=p^2$, then $F$ is not a simple extension of $K$. Exhibit an infinite number of intermediate fields. Thank you.
Suppose $F=K(\alpha)$. Write $\alpha$ as a rational function in $u$ and $v$. What can you say about $\alpha^p$? Why is this a contradiction?