Characteristic (algebra)

In mathematics, the characteristic of a ring R with identity element 1_R is defined to be the smallest positive integer n such that n1_R = 0 (where n1_R is defined as 1_R + ... + 1_R with n summands). If no such n exists, we say that the characteristic of R is 0. The characteristic of R is often denoted char(R).

The characteristic of the ring R may be equivalently defined as the unique natural number n such that nZ is the kernel of the unique ring homomorphism from Z to R which sends 1 to 1_R. And yet another equivalent definition: the characteristic of R is the 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.
• The only ring with characteristic 1 is the trivial ring which has only a single element 0=1.
• If the non-trivial ring R does not have any 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.
• 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 ring of characteristic 0 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 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 = p^nm.)
• If a commutative ring 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.

External links

• Finite fields - Wikibook link.