Characteristic (algebra)

In a field of a ring, the smallest positive integer, if any, such that the sum of n ones equals 0; zero otherwise

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero.

The special definition of the characteristic zero is motivated by the equivalent definitions given in § Other equivalent characterizations, 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 n such that:[2]

for every element a of the ring (again, if n exists; otherwise zero). Some authors do not include the multiplicative identity element in their requirements for a ring (see Multiplicative identity: mandatory vs. optional), and this definition is suitable for that convention; otherwise the two definitions are equivalent due to the distributive law in rings.

If R and S are rings and there exists a ring homomorphism RS, then the characteristic of S divides the characteristic of R. This can sometimes be used to exclude the possibility of certain ring homomorphisms. The only ring with characteristic 1 is the zero ring, which has only a single element 0 = 1. If a nontrivial ring R does not have any nontrivial zero divisors, then its characteristic is either 0 or prime. In particular, this applies to all fields, to all integral domains, and to all division rings. Any ring of characteristic 0 is infinite.

The ring Z/nZ of integers modulo n has characteristic n. If R is a subring of S, then R and S have the same characteristic. For example, if p is prime and q(X) is an irreducible polynomial with coefficients in the field Fp, then the quotient ring Fp[X] / (q(X)) is a field of characteristic p. Another example: The field C of complex numbers contains Z, so the characteristic of C is 0.

A Z/nZ-algebra is equivalently a ring whose characteristic divides n. This is because for every ring R there is a ring homomorphism ZR, and this map factors through Z/nZ 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 "freshman's dream" holds for power p. The map f(x) = xp then defines a ring homomorphism RR. It is called the Frobenius homomorphism. If R is an integral domain it is injective.

As mentioned above, the characteristic of any field is either 0 or a prime number. A field of non-zero characteristic is called a field of finite characteristic or positive characteristic or prime characteristic.

Any field F has a unique minimal subfield, also called its prime field. This subfield is isomorphic to either the rational number field Q or a finite field Fp of prime order. Two prime fields of the same characteristic are isomorphic, and this isomorphic is unique. In other words, there is essentially a unique prime field in each characteristic. The most common fields of characteristic zero that are the subfields of the complex numbers. The p-adic fields are characteristic zero fields that are widely used in number theory. They have absolute values which are very different from those of complex numbers.

For any ordered field, as the field of rational numbers Q or the field of real numbers R, the characteristic is 0. Thus, number fields and the field of complex numbers C are of characteristic zero. Actually, every field of characteristic zero is the quotient field of a ring Q[X]/P where X is a set of variables and P a set of polynomials in Q[X]. The finite field GF(pn) has characteristic p. There exist infinite fields of prime characteristic. For example, the field of all rational functions over Z/pZ, the algebraic closure of Z/pZ or the field of formal Laurent series Z/pZ((T)). The characteristic exponent is defined similarly, except that it is equal to 1 if the characteristic is zero; otherwise it has the same value as the characteristic.[3]

The size of any finite ring of prime characteristic p is a power of p. Since in that case it must contain Z/pZ it must also be a vector space over that field and from linear algebra we know that the sizes of finite vector spaces over finite fields are a power of the size of the field. This also shows that the size of any finite vector space is a prime power. (It is a vector space over a finite field, which we have shown to be of size pn, so its size is (pn)m = pnm.)