# Characteristic of Integral Domain is Zero or Prime

## Theorem

Let $\struct {D, +, \circ}$ be an integral domain.

Let $\operatorname{Char} \left({D}\right)$ be the characteristic of $D$.

Then $\operatorname{Char} \left({D}\right)$ is either $0$ or a prime number.

## Proof

By definition, an integral domain has no proper zero divisors.

If $\struct {D, +, \circ}$ is finite, then from Characteristic of Finite Ring with No Zero Divisors, $\operatorname{Char} \left({D}\right)$ is prime.

On the other hand, suppose $\struct {D, +, \circ}$ is not finite.

Then there are no $x, y \in D, x \ne 0 \ne y$ such that $x + y = 0$.

Thus it follows that $\operatorname{Char} \left({D}\right)$ is zero.

$\blacksquare$