Showing an element in a Finite Field can be written as a power. I had a question that I'm stuck with:
Show that every element in $GF(p^n)$ can be written in the form of $a^p$ for some unique $a\in GF(p^n)$.
So this field is the splitting field for the polynomial $f(a) = a^{p^n} - a$ which is what I can understand, but I really don't have any idea how to progress further with the problem.

This is the Frobenius automorphism $\phi:K \rightarrow K$, where $K$ is some finite field extension of $GF(p)$.  
Hint: Show this map is injective by showing that $ker(\phi)={{0}}$.  This will instantly imply that $\phi$ is an isomorphism, and hence that every element of $K (=GF(p^n))$ can be written as $a^p$ for some $a \in K$.
