Galois Field has Non-Zero Characteristic/Proof 2

Proof
Let $\GF$ be a Galois field.

Let $P$ be its prime subfield.

Suppose $\Char \GF = 0$.

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.