# Galois Field is Perfect

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## 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.

$\blacksquare$