Characteristic of Galois Field is Prime

Theorem
Let $F$ be a finite field.

Then the characteristic of $F$ is a prime number.

Proof
By Characteristic of Field is Zero or Prime, it follows that $\operatorname{Char} \left({F}\right)$ is $0$ or a prime number.

$F$ is by definition a finite ring, so by Characteristic of Finite Ring is Non-Zero, $\operatorname{Char} \left({F}\right) \ne 0$.

Thus $\operatorname{Char} \left({F}\right)$ is a prime number.