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# Characteristic

In abstract algebra, the characteristic of a ring R is defined to be the smallest positive integer n such that 1R+...+1R (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 = pnm.)
• If an integral domain R has prime characteristic p, then we have (x + y)p = xp + yp for all elements x and y in R. The map f(x) = xp 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'.