I am reading through Dummit & Foote chapters 13 and 14 trying to find something that will help with questions like this, but I still don't have any tools:
Calculate $[F_{p^2}(t)(t^{1/p}):F_p(t)]$
If I try to think of this as calculating the basis of the vector space $F_{p^2}(t)(t^{1/p})$ over $F_p(t)$ I am lost. I know basic things about degrees of fields, like $[F_{p^2}:F_p]=2$, but that's a far cry from being able to calculate degrees of rational functions of rational functions over rational functions. Any suggestions?
One idea I have is to think of this as: $F_p \subseteq F_{p^2}\subseteq F_{p^2}(t)\subseteq F_{p^2}(t)(t^{1/p})$, though I'm not sure that quite makes sense, since $F_{p^2}$ is the splitting field of $x^{p^2}-1$ over $F_p$ and I'm not sure that it contains $F_p$.
Think of it as $F_p(t) \subset F_{p^2}(t) \subset F_{p^2}(t^{1/p})$, which both have finite degree.