characteristic

Let $(F,+,\cdot )$ be a field. The characteristic $\mathrm{Char}(F)$ 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, $\mathrm{Char}(F)$ is defined to be $0$.

• if $f:\mathbb{Z}\to F$ is defined by $f(n)=n\cdot 1$ then $\mathrm{Char}(F)$ is the least strictly positive generator of $\ker (f)$ if $\ker (f)\ne \{0\}$; otherwise it is $0$.

• if $K$ is the prime subfield of $F$, then $\mathrm{Char}(F)$ 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.