Galois Field is Perfect

Theorem
Let $F$ be a finite field.

Then $F$ is perfect.

Proof
By Characteristic of Finite Field is Prime, $\operatorname{char}(F)$ is a prime number, say $p$.

By Frobenius Endomorphism on Field is Injective, $\operatorname{Frob}$ is injective.

By Injection from Finite Set to Itself is Surjection, $\operatorname{Frob}$ is bijective.

By Bijective Ring Homomorphism is Isomorphism, $\operatorname{Frob}$ is an automorphism.