# Galois Field has Non-Zero Characteristic

(Redirected from Finite 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.

$\blacksquare$

## 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.

$\blacksquare$

## Sources

- 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): $\S 1.3$: Theorem $3.2$: Corollary