Galois Field is Perfect

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Theorem

Let $F$ be a Galois field.


Then $F$ is perfect.


Proof

By Characteristic of Galois 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.

$\blacksquare$