Let $p$ be a prime and $F$ a finite field of $p^n$ elements. Suppose $F^\times=\langle x \rangle$. Let $\phi$ be an automorhpism of $F$. Then prove that $\phi(x)=x^{p^r}$ for some integer $r$. How to prove? Thanks.
If you use theorems in Galois theory, could you give citations (which page on which book or website) for me to study?