Galois Field is Perfect

Theorem

Let $\GF$ be a Galois field.

Then $\GF$ is perfect.

Proof

By Characteristic of Galois Field is Prime, $\Char \GF$ is a prime number, say $p$.

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

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

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

$\blacksquare$