I wish to prove the following claim:
Let $\mathbb{F}$ be a finite field of characteristic $p$ then the map $a\to a^p$ is surjective
Dummit and Foote's Abstract Algebra says that this map is injective hence surjective, but isn't this is just an application of Fermat's little theorem ?
Best Answer
No, Fermat's little theorem is about $\mathbb F_p$ and you need to it for $\mathbb F_q$, where $q$ is a power of $p$.
But you don't need to know that $\mathbb F$ has order a power of $p$, just that it is finite and that every injective map is surjective, as the book says.