How do I complete the proof that the order of the automorphism $x \mapsto x^{p}$ of $\mathbb{F}_{p^{n}}$ has order $n$?

Question

I want to prove that the order of the automorphism $x \mapsto x^{p}$ of $\mathbb{F}_{p^{n}}$ is $n$.

What I have so far shows is $\varphi^{n}(\alpha)=\alpha$ holds for all $\alpha \in \mathbb{F}_{p^{n}}$ so that $\varphi^{n}=1$.

If on the other hand $\varphi^{t}=1$, then for all $\alpha \in \mathbb{F}_{p^{n}}$ we have $\alpha^{p^{t}}-\alpha=0$, so that each $\alpha$ satisfies $x^{p^{t}}-x$. I'm not sure how to move from here to conclude $n \le t$.

Answer 1

If all elements of $\Bbb F_{p^n}$ satisfy a given polynomial, then that polynomial has $p^n$ distinct roots, so its degree must be at least $p^n$.