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, 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.
- 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(p^{n}) 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 p^{n}. So its size is (p^{n})^{m} = p^{nm}. QED)
- 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'.
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