Characteristic of a field

2020 Mathematics Subject Classification: Primary: 12Exx [MSN][ZBL]

An invariant of a field which is either a prime number or the number zero, uniquely determined for a given field in the following way. If for some $n > 0$, $$ 0 = ne = \underbrace{e+e+\cdots+e}_{n\,\text{summands}} $$ where $e$ is the unit element of the field $F$, then the smallest such $n$ is a prime number; it is called the characteristic of $F$. If there are no such numbers $n$, then one says that the characteristic of $F$ is zero or that $F$ is a field of characteristic zero. Sometimes such a field is said to be without characteristic or of characteristic infinity $(\infty)$. Every field of characteristic zero contains a subfield isomorphic to the field of all rational numbers, and a field of finite characteristic $p$ contains a subfield isomorphic to the field of residue classes modulo $p$: in each case this is the prime field.