In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest positive number of copies of the ring's multiplicative identity (1) that will sum to the additive identity (0).
If no such number exists, the ring is said to have characteristic zero.
That is, char(R) is the smallest positive number n such that:[1](p 198, Thm.
The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately.
The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive integer n such that:[1](p 198, Def.
This definition applies in the more general class of rngs (see Ring (mathematics) § Multiplicative identity and the term "ring"); for (unital) rings the two definitions are equivalent due to their distributive law.
In particular, this applies to all fields, to all integral domains, and to all division rings.
For example, if p is prime and q(X) is an irreducible polynomial with coefficients in the field
if and only if the characteristic of R divides n. In this case for any r in the ring, then adding r to itself n times gives nr = 0.
If a commutative ring R has prime characteristic p, then we have (x + y)p = xp + yp for all elements x and y in R – the normally incorrect "freshman's dream" holds for power p. The map x ↦ xp then defines a ring homomorphism R → R, which is called the Frobenius homomorphism.
As mentioned above, the characteristic of any field is either 0 or a prime number.
This subfield is isomorphic to either the rational number field
In other words, there is essentially a unique prime field in each characteristic.
The most common fields of characteristic zero are the subfields of the complex numbers.
They have absolute values which are very different from those of complex numbers.
The finite field GF(pn) has characteristic p. There exist infinite fields of prime characteristic.
or the field of formal Laurent series
The size of any finite ring of prime characteristic p is a power of p. Since in that case it contains
This also shows that the size of any finite vector space is a prime power.