Characteristic (algebra)

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In mathematics, the characteristic of a ring R is defined to be the smallest positive integer n such that

n a = 0, for all a in R.

where na is defined as

a + ... + a with n summands.

If no such n exists, the characteristic of R is by definition 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.

[edit] The case of rings

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. 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 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.

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

then defines a ring homomorphism

R → R.

It is called the Frobenius homomorphism. If R is an integral domain it is injective.

[edit] The case of fields

As mentioned above, the characteristic of any field is either 0 or a prime number.

For any field F, there is a minimal subfield, namely the prime field, the smallest subfield containing 1_F. It is isomorphic either to the rational number field Q, or a finite field; the structure of the prime field and the characteristic each determine the other. Fields of characteristic zero have the most familiar properties; for practical purposes they resemble subfields of the complex numbers (unless they have very large cardinality, that is). The p-adic fields are characteristic zero fields, much applied in number theory, that are constructed from rings of characteristic p^k, as k → ∞.

For any ordered field (for example, the rationals or the reals) the characteristic is 0. 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.)