Characteristic

In abstract algebra, the characteristic of a ring R is defined to be the smallest positive integer n such that 1_R+...+1_R (with n summands) yields 0. If no such n exists, we say that the characteristic of R is 0.

Alternatively and equivalently, the characteristic of the ring R may be defined as that unique natural number n such that R contains a subring isomorphic to the factor ring Z/nZ.

Examples and notes:

If R and S are rings and there exists a ring homomorphism R -> S, then the characteristic of S divides the characteristic of R. This can sometimes be used to exclude the possibility of certain ring homomorphisms.
For any integral domain (and in particular for any field), the characteristic is either 0 or prime.
For any ordered field (for example, the rationals or the reals) the characteristic is 0.
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 instance, if q(X) is a prime polynomial with coefficients in the field Z/pZ where p is prime, then the factor ring (Z/pZ)[X]/(q(X)) is a field of characteristic p. Since the complex numbers contain the rationals, their characteristic is 0.
Any field of 0 characteristic is infinite. 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 is one such. The algebraic closure of Z/pZ is another example.
The size of any finite field of 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 = p^nm.)
If an integral domain R has prime characteristic p, then we have (x + y)^p = x^p + y^p for all elements x and y in R. The map f(x) = x^p defines an injective ring homomorphism R -> R. It is called the Frobenius homomorphism.

Characteristic is also sometimes used as a piece of jargon in discussions of universals in metaphysics, often in the phrase 'distinguishing characteristics'.