View source for Characteristic of a field

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{{MSC|12Exx}}

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]].