In the finite field $F$ of characteristic $p$, is $a^{p^n} = a$?

Question

If F is a finite field of characteristic $p$, $a$ is some element in $F$ and the number of elements in $F$ is $p^n$, is it true that $a^{p^n} = a$ for all $a$ in $F$? If it is, how could one prove or motivate that?

Answer 1

$F-0$ is a group of order $p^n-1$, so $x^{p^n-1}=1$ if $x\neq 0$. This implies that $x^{p^n}=xx^{p^n-1}=x$ if $x\neq 0$. the result. Obviously, $x^{p^n}=x$ if $x=0$.

Answer 2

Hint: look at the multiplicative group.

Answer 3

Every finite subgroup of the multiplicative group of a field is cyclic. As the multiplicative group of a field of order $p^n$ is finite of order $p^n-1$ the result follows.