Finite Fields are Perfect

Theorem.
Every finite field $\mathbb{F}_q$ is perfect . In characteristic $p$, this is equivalent to the Frobenius map $x\mapsto x^p$ being an automorphism (i.e., bijective). Consequently, every algebraic extension of a finite field is separable .

Examples.

1. In $\mathbb{F}_{p^n}$, the map $x\mapsto x^p$ is injective because the field has no zero divisors, hence bijective because the set is finite.
2. The polynomial $x^{p^n}-x$ has derivative $-1$, so its roots in $\mathbb{F}_{p^n}$ are all distinct (see distinct roots criterion ).
3. Any finite extension $\mathbb{F}_{p^n}/\mathbb{F}_p$ is separable (an instance of perfect base ⇒ separable ).

Related. Galois groups of finite fields .