Let (F,+,) be a field. The characteristicPlanetmathPlanetmath Char(F) of F is commonly given by one of three equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath definitions:

  • if there is some positive integer n for which the result of adding any elementMathworldMathworld to itself n times yields 0, then the characteristic of the field is the least such n. Otherwise, Char(F) is defined to be 0.

  • if f:F is defined by f(n)=n1 then Char(F) is the least strictly positive generator of ker(f) if ker(f){0}; otherwise it is 0.

  • if K is the prime subfieldMathworldPlanetmath of F, then 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.

Title characteristic
Canonical name Characteristic
Date of creation 2013-03-22 12:05:01
Last modified on 2013-03-22 12:05:01
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 16
Author Mathprof (13753)
Entry type Definition
Classification msc 12E99