Find the # of elements of $F$ that can be written in the form $a^n-a$.

Question

Im trying to show that for a finite field $F$ or order $ p^n $ there are $p^{n-1}$ elements that can be written in the form $a^p-a$ for some $a\in F$

I know that if we consider $x^p-x$, then $D_xf(x)=px^{p-1}-1=-1$ since the characteristic of $F$ is $p$. So then $x^p-x$ is separable and has $p$ distinct $p$-th roots of unity. Im not sure how to continue from here.

Answer 1

Hint: Considering $F$ as a group under addition, show that the function $f:F\to F$ defined by $f(a)=a^p-a$ is a homomorphism of groups. Then use the first isomorphism theorem.

Answer 2

Equally, you could view $f(x) = x^p - x$ as a linear mapping. You then need to find the kernel of $f$. $f$ has at most $p$ roots, because it has degree $p$. $f$ in fact has exactly $p$ roots, which lie in a 1d subspace. Finally, use the nullity-rank theorem to conclude that the rank of $f$ is $n - 1$.