In mathematics, the characteristic of a ring R with identity element 1_R is defined to be the smallest positive integer n such that n1_R = 0 (where n1_R is defined as 1_R + ... + 1_R with n summands). If no such n exists, we say that the characteristic of R is 0. The characteristic of R is often denoted char(R). History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ...

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. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ... Given a ring (R, +, *), we say that a subset S of R is a subring thereof if it is a ring under the restriction of + and * thereto, and contains the same unity as R. A subring is just a subgroup of (R, +) which contains the identity and is closed under... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ...

The case of rings

If R and S are rings and there exists a ring homomorphism In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ...

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. In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ... In mathematics, a prime number, or prime for short, is a natural number greater than one and whose only distinct positive divisors are one and itself. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero. ... In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. ...

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. Modular arithmetic is a system of arithmetic for integers, sometimes referred to as clock arithmetic, where numbers wrap around after they reach a certain value (the modulus). ... Given a ring (R, +, *), we say that a subset S of R is a subring thereof if it is a ring under the restriction of + and * thereto, and contains the same unity as R. A subring is just a subgroup of (R, +) which contains the identity and is closed under... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

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 defines an injective ring homomorphism R → R. It is called the Frobenius homomorphism. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ...

The case of fields

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. In mathematics, an ordered field is a field (F,+,*) together with a total order ≤ on F that is compatible with the algebraic operations in the following sense: if a ≤ b then a + c ≤ b + c if 0 ≤ a and 0 ≤ b then 0 ≤ a b It follows from these axioms... In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... The text or formatting below is generated by a template which has been proposed for deletion. ... In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ... In mathematics, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ...

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.) The fundamental concept in linear algebra is that of a vector space or linear space. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ...

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. In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

External links

• Finite fields - Wikibook link.