[[Integral domain]]
# The characteristic of an integral domain is 0 or prime

Let $D$ be an [[integral domain]]. Then its [[characteristic]] $\mathrm{char}(D)$ is either 0 or prime. #m/thm/ring

> [!check]- Proof
> Suppose that $1$ has order $n = st$ where $1 \leq s,t \leq n$.
> Then
> $$
> \begin{align*}
> 0 = \sum^n 1 = \left( \sum^{s} 1 \right) \left( \sum^t 1 \right)
> \end{align*}
> $$
> so either $s = n$ or $t = n$.
> Thus $n$ is prime.
> <span class="QED"/>

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