characteristic

Let $(F,+,\cdot)$ be a field. The characteristic $\cha(F)$ of $F$ is commonly given by one of three equivalent definitions:

• if there is some positive integer $n$ for which the result of adding any element to itself $n$ times yields $0$ , then the characteristic of the field is the least such $n$ . Otherwise, $\cha(F)$ is defined to be $0$ .
• if $f:\mathbb{Z}\to F$ is defined by $f(n) = n\cdot 1$ then $\cha(F)$ is the least strictly positive generator of $\operatorname{ker}(f)$ if $\operatorname{ker}(f)\neq \{ 0\}$ ; otherwise it is $0$ .
• if $K$ is the prime subfield of $F$ , then $\cha(F)$ is the size of $K$ if this is finite, and $0$ otherwise.

Note that the first definition also applies to arbitrary rings, and not just to fields.

The characteristic of a field (or more generally an integral domain) is always prime. For if the characteristic of $F$ were composite, say $mn$ for $m,n>1$ , then in particular $mn$ would equal zero. Then either $m$ would be zero or $n$ would be zero, so the characteristic of $F$ would actually be smaller than $mn$ , contradicting the minimality condition.