Characteristic of an integral domain is 0 or prime

Characteristic of an integral domain is 0 or prime: If $D$ is an integral domain, then $\operatorname{char}(D)$ is either $0$ or a prime number $p$.

The proof analyzes the kernel of the canonical map $\mathbb Z\to D$ using the kernel–ideal property , and thereby determines the characteristic of an integral domain . This dichotomy is the starting point for constructing the smallest subfield inside any field (see the corollary on prime subfields).