# 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

*n*1_{R}= 0,

where *n*1_{R} is defined as

- 1
_{R}+ ... + 1_{R}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 *n***Z** 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**/*n***Z**.

## 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**/*n***Z** 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**/*p***Z** where *p* is prime, then the factor ring (**Z**/*p***Z**)[*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.

## The case of fields

For any field *F*, there is a minimal subfield, namely the **prime field**, the smallest subfield containing 1_{F</sup>. 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 pk, as k → ∞.
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For any ordered field (for example, the rationals or the reals) the characteristic is 0. 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**/*p***Z** is one such. The algebraic closure of **Z**/*p***Z** 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**/*p***Z** 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}.)

## External links

- Finite fields - Wikibook link.