The number of solution to the equation $x^p=a$ is either $0$ or $p-1$ in a field of charestrics $p$.

Question

Let $a$ be a non-zero element of $GF(p^n)$. Prove that the number of solutions of $x^{p-1} = a$ is $0$ or $p−1$.

If it would have $\mathbb{F}_p$ then I was able to prove that given any non-zero $a\in \mathbb{F}_p,\ x^{p-1}=a$ has $p-1$ solutions. But I ma unable to do this.

Answer 1

First, we can reduce it to the case when $a=1$: if $b\in\mathrm{GF}\left(p^n\right)$ is a solution to $x^{p-1}=a$, then the set of solutions of this equation is $$\left\{b\cdot\rho\mid\rho^{p-1}=1\right\}.$$

Now, the equation $x^{p-1}=1$ has exactly $p-1$ roots, which are the elements of $\mathbb{F}_p^{\times}$ (can you see why?).

This solves the question.