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

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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 $0$.
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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.
Title  characteristic 

Canonical name  Characteristic 
Date of creation  20130322 12:05:01 
Last modified on  20130322 12:05:01 
Owner  Mathprof (13753) 
Last modified by  Mathprof (13753) 
Numerical id  16 
Author  Mathprof (13753) 
Entry type  Definition 
Classification  msc 12E99 