## CHARACTERISTIC

### Characteristic

In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this sum never reaches the additive identity.

The above text is a snippet from Wikipedia: Characteristic (algebra)

### characteristic

#### Noun

1. a distinguishable feature of a person or thing
2. the integer part of a logarithm
3. the distinguishing features of a navigational light on a lighthouse etc by which it can be identified (colour, pattern of flashes etc)
4. The minimum number of times that the unit of a field must be added unto itself in order to yield that field's zero, or, if that minimum natural number does not exist, then (the integer) zero.
A field's characteristic, if non-trivial, must be prime, otherwise the field would not be an integral domain, which fields must be. For example, if a field's characteristic were six, then (1+1)+(1+1)+(1+1) = 0; if 1+1 were labeled A and 1+1+1 were labeled B, then A + A + A = 0, A (1 + 1 + 1) = 0 by distributivity; A B = 0, and the field could not be an integral domain.