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Encyclopedia > Characteristic (algebra)
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In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... 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. ... Jump to: navigation, search The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

n1R = 0,

where n1R is defined as

1R + ... + 1R 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 1R. 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. Jump to: navigation, search 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), and they can be used... 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. ...

RS,

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. ... Jump to: navigation, search In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... Jump to: navigation, search 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 same 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. Jump to: navigation, search Modular arithmetic is a system of arithmetic for integers, 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... Jump to: navigation, search In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... Jump to: navigation, search In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...


If a commutative ring R has prime characteristic p, then we have (x + y)p = xp + yp for all elements x and y in R.


The map

f(x) = xp

then defines a ring homomorphism In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ...

RR.

It is called the Frobenius homomorphism. If R is an integral domain it is injective. Jump to: navigation, search In mathematics, the Frobenius morphism is typically viewed as a ring endomorphism of a commutative ring of prime characteristic given by . There are extensions of this definition to module over rings of prime characteristic. ... 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 mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...


The case of fields

For any field F, there is a minimal subfield, namely the prime field, the smallest subfield containing 1F. 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 → ∞. Jump to: navigation, search In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... Jump to: navigation, search In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... The title given to this article is incorrect due to technical limitations. ...


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... Jump to: navigation, search In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... 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, a rational function is a ratio of polynomials. ... In mathematics, particularly abstract algebra, 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 = pnm.) Jump to: navigation, search A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... Jump to: navigation, search 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. ...


External links

  • Finite fields - Wikibook link.

  Results from FactBites:
 
Characteristic (algebra) - definition of Characteristic (algebra) in Encyclopedia (501 words)
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
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.
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.
  More results at FactBites »

 
 

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