Galois Field has Non-Zero Characteristic/Proof 2

Theorem

Let $\GF$ be a Galois field.

Then the characteristic of $\GF$ is non-zero.

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.

$\blacksquare$

Sources

• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 3$. Homomorphisms: Theorem $3.2$: Corollary