Characteristic (algebra)

Template:Short description In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest positive number of copies of the ring's multiplicative identity (1) that will sum to the additive identity (0). If no such number exists, the ring is said to have characteristic zero.

That is, char(R) is the smallest positive number n such that:Template:Rp

$\underset{n\text{ summands}}{\underbrace{1+\cdots +1}}=0$

if such a number n exists, and 0 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 n such that:Template:Rp

$\underset{n\text{ summands}}{\underbrace{a+\cdots +a}}=0$

for every element a of the ring (again, if n exists; otherwise zero). This definition is equivalent for a ring, because of distributivity. For rngs (rings without identity), the former definition is nonsensical, and the latter definition is generally used.

Integers, rational numbers and real numbers have characteristic 0.

The integers modulo n have characteristic n.

Every Boolean ring has characteristic 2.

The characteristic of a field is either 0 or a prime number.

Equivalent characterizations

• The characteristic of a ring R is the natural number n such that n$\mathbb{\mathbb{Z}}$ is the kernel of the unique ring homomorphism from $\mathbb{\mathbb{Z}}$ to R.Template:Efn
• The characteristic is the natural number n such that R contains a subring isomorphic to the factor ring $\mathbb{\mathbb{Z}}/n\mathbb{\mathbb{Z}}$, which is the image of the above homomorphism.
• When the non-negative integers Template:Mset are partially ordered by divisibility, then 1 is the smallest and 0 is the largest. Then the characteristic of a ring is the smallest value of n for which n ⋅ 1 = 0. If nothing "smaller" (in this ordering) than 0 will suffice, then the characteristic is 0. This is the appropriate partial ordering because of such facts as that char(A × B) is the least common multiple of char A and char B, and that no ring homomorphism f : A → B exists unless char B divides char A.
• The characteristic of a ring R is n precisely if the statement ka = 0 for all a ∈ R implies that k is a multiple of n.

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 zero ring, which has only a single element 0. If a nontrivial ring R does not have any nontrivial 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 zero is infinite.

The ring $\mathbb{\mathbb{Z}}/n\mathbb{\mathbb{Z}}$ of integers modulo n has characteristic n. If R is a subring of S, then R and S have the same characteristic. For example, if p is prime and q(X) is an irreducible polynomial with coefficients in the field ${\mathbb{𝔽}}_p$ with Template:Mvar elements, then the quotient ring ${\mathbb{𝔽}}_p[X]/(q(X))$ is a field of characteristic p. Another example: The field $\mathbb{ℂ}$ of complex numbers contains $\mathbb{\mathbb{Z}}$, so the characteristic of $\mathbb{ℂ}$ is 0.

A $\mathbb{\mathbb{Z}}/n\mathbb{\mathbb{Z}}$-algebra is equivalently a ring whose characteristic divides n. This is because for every ring R there is a ring homomorphism $\mathbb{\mathbb{Z}}\to R$, and this map factors through $\mathbb{\mathbb{Z}}/n\mathbb{\mathbb{Z}}$ if and only if the characteristic of R divides n. In this case for any r in the ring, then adding r to itself n times gives nr = 0.

If a commutative ring R has prime characteristic p, then we have (x + y)Template:I sup = xTemplate:I sup + yTemplate:I sup for all elements x and y in R – the normally incorrect "freshman's dream" holds for power p. The map x ↦ xTemplate:I sup then defines a ring homomorphism R → R, which is called the Frobenius homomorphism. If R is also an integral domain, the homomorphism is injective.

Case of fields

As mentioned above, the characteristic of any field is either 0 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 1 when the characteristic is 0; otherwise it has the same value as the characteristic.

Any field F has a unique minimal subfield, also called its prime field. This subfield is isomorphic to either the rational number field $\mathbb{\mathbb{Q}}$ or a finite field ${\mathbb{𝔽}}_p$ 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 $\mathbb{ℝ}$ 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 $\mathbb{\mathbb{Q}}$ and $\mathbb{ℝ}.$

Fields of prime characteristic

The finite field GF(pⁿ) has characteristic p.

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

The size of any finite ring of prime characteristic p is a power of p. Since in that case it contains $\mathbb{\mathbb{Z}}/p\mathbb{\mathbb{Z}}$ 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