Characteristic of Galois Field is Prime

Theorem
Let $\F$ be a Galois field.

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

Proof
By Characteristic of Field is Zero or Prime, it follows that $\Char \F$ is $0$ or a prime number.

By Finite Field has Non-Zero Characteristic:
 * $\Char \F \ne 0$

Thus $\Char \F$ is a prime number.