Characteristic (algebra)

In mathematics, the characteristic of a ring Template:Math, often denoted Template:Math, is defined to be the smallest positive number of copies of the ring's multiplicative identity (Template:Math) that will sum to the additive identity (Template:Math). If no such number exists, the ring is said to have characteristic zero.

That is, Template:Math is the smallest positive number Template:Math such that:<ref name=Fraleigh-Brand-2020/>Template:Rp

<math>\underbrace{1+\cdots+1}_{n \text{ summands}} = 0</math>

if such a number Template:Math exists, and Template:Math otherwise.

Motivation

The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately.

The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive integer Template:Math such that:<ref name=Fraleigh-Brand-2020> Template:Cite book </ref>Template:Rp

<math>\underbrace{a+\cdots+a}_{n \text{ summands}} = 0</math>

for every element Template:Math of the ring (again, if Template:Math exists; otherwise zero). This definition applies in the more general class of rngs (see Template:Slink); for (unital) rings the two definitions are equivalent due to their distributive law.

Equivalent characterizations

• The characteristic of a ring Template:Math is the natural number Template:Math such that Template:Math is the kernel of the unique ring homomorphism from <math>\mathbb{Z}</math> to Template:Math.Template:Efn
• The characteristic is the natural number Template:Math such that Template:Math contains a subring isomorphic to the factor ring <math>\mathbb{Z}/n\mathbb{Z}</math>, which is the image of the above homomorphism.
• When the non-negative integers Template:Math are partially ordered by divisibility, then Template:Math is the smallest and Template:Math is the largest. Then the characteristic of a ring is the smallest value of Template:Math for which Template:Math. If nothing "smaller" (in this ordering) than Template:Math will suffice, then the characteristic is Template:Math. This is the appropriate partial ordering because of such facts as that Template:Math is the least common multiple of Template:Math and Template:Math, and that no ring homomorphism Template:Math exists unless Template:Math divides Template:Math.
• The characteristic of a ring Template:Math is Template:Math precisely if the statement Template:Math for all Template:Math implies that Template:Math is a multiple of Template:Math.

Case of rings

If Template:Math and Template:Math are rings and there exists a ring homomorphism Template:Math, then the characteristic of Template:Math divides the characteristic of Template:Math. This can sometimes be used to exclude the possibility of certain ring homomorphisms. The only ring with characteristic Template:Math is the zero ring, which has only a single element Template:Math. If a nontrivial ring Template:Math does not have any nontrivial zero divisors, then its characteristic is either Template:Math or prime. In particular, this applies to all fields, to all integral domains, and to all division rings. Any ring of characteristic zero is infinite.

The ring <math>\mathbb{Z}/n\mathbb{Z}</math> of integers modulo Template:Math has characteristic Template:Math. If Template:Math is a subring of Template:Math, then Template:Math and Template:Math have the same characteristic. For example, if Template:Math is prime and Template:Math is an irreducible polynomial with coefficients in the field <math>\mathbb F_p</math> with Template:Mvar elements, then the quotient ring <math>\mathbb F_p[X]/(q(X))</math> is a field of characteristic Template:Math. Another example: The field <math>\mathbb{C}</math> of complex numbers contains <math>\mathbb{Z}</math>, so the characteristic of <math>\mathbb{C}</math> is Template:Math.

A <math>\mathbb{Z}/n\mathbb{Z}</math>-algebra is equivalently a ring whose characteristic divides Template:Math. This is because for every ring Template:Math there is a ring homomorphism <math>\mathbb{Z}\to R</math>, and this map factors through <math>\mathbb{Z}/n\mathbb{Z}</math> if and only if the characteristic of Template:Math divides Template:Math. In this case for any Template:Math in the ring, then adding Template:Math to itself Template:Math times gives Template:Math.

If a commutative ring Template:Math has prime characteristic Template:Math, then we have Template:Math for all elements Template:Math and Template:Math in Template:Math – the normally incorrect "freshman's dream" holds for power Template:Math. The map Template:Math then defines a ring homomorphism Template:Math, which is called the Frobenius homomorphism. If Template:Math is an integral domain it is injective.

Case of fields

As mentioned above, the characteristic of any field is either Template:Math or a prime number. A field of non-zero characteristic is called a field of finite characteristic or positive characteristic or prime characteristic. The characteristic exponent is defined similarly, except that it is equal to Template:Math when the characteristic is Template:Math; otherwise it has the same value as the characteristic.<ref> Template:Cite book</ref>

Any field Template:Math has a unique minimal subfield, also called its prime field. This subfield is isomorphic to either the rational number field <math>\mathbb{Q}</math> or a finite field <math>\mathbb F_p</math> of prime order. Two prime fields of the same characteristic are isomorphic, and this isomorphism is unique. In other words, there is essentially a unique prime field in each characteristic.

Fields of characteristic zero

The fields of characteristic zero are those that have a subfield isomorphic to the field Template:Tmath of the rational numbers. The most common of such fields are the subfields of the field Template:Tmath of the complex numbers; this includes the real numbers <math>\mathbb{R}</math> and all algebraic number fields.

Other fields of characteristic zero are the p-adic fields that are widely used in number theory.

Fields of rational fractions over the integers or a field of characteristic zero are other common examples.

Ordered fields always have characteristic zero; they include <math>\mathbb{Q}</math> and <math>\mathbb{R}.</math>

Fields of prime characteristic

The finite field Template:Math has characteristic Template:Math.

There exist infinite fields of prime characteristic. For example, the field of all rational functions over <math>\mathbb{Z}/p\mathbb{Z}</math>, the algebraic closure of <math>\mathbb{Z}/p\mathbb{Z}</math> or the field of formal Laurent series <math>\mathbb{Z}/p\mathbb{Z}((T))</math>.

The size of any finite ring of prime characteristic Template:Math is a power of Template:Math. Since in that case it contains <math>\mathbb{Z}/p\mathbb{Z}</math> it is also 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.Template:Efn

See also

• Ring of mixed characteristic

Sources

• Template:Cite book