characteristic
Let (F,+,⋅) be a field. The characteristic 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, Char(F) is defined to be 0.
- •
-
•
if K is the prime subfield
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 |