Let $F$ be a field of characteristic p. Is field extension $F/F^p$ finite?

Question

I have a field $F$ with characteristic $p$. Let's define $F^p = \{a^p \mid a \in F\}$. And we have the field extension $F/F^p$. I see how this extension is algebraic, but can it be an extension of a finite degree? How do I prove it if it's true? I don't really see how it can be finite if the field $F$ is infinite.

Thank you