Let be a field. The characteristic of is commonly given by one of three equivalent definitions:

• if there is some positive integer for which the result of adding any element to itself times yields 0, then the characteristic of the field is the least such . Otherwise, is defined to be 0.
• if is defined by then is the least strictly positive generator of if ; otherwise it is 0.
• if is the prime subfield of , then is the size of 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 were composite, say for , then in particular would equal zero. Then either would be zero or would be zero, so the characteristic of would actually be smaller than , contradicting the minimality condition.