Galois Field has Non-Zero Characteristic

Theorem
Let $F$ be a Galois field.

Then the characteristic of $F$ is non-zero.

Proof 1
A direct application of Characteristic of Finite Ring is Non-Zero.

Proof 2
Let $F$ be a Galois field.

Let $P$ be its prime subfield.

Suppose $\operatorname {Char} \left({F}\right) = 0$.

Then from Field of Characteristic Zero has Unique Prime Subfield, $P$ is isomorphic to $\Q$ which is infinite.

But a Galois field can not have an infinite subfield.