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Field (mathematics)

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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.

Fields are important objects of study in algebra since they provide the proper generalization of number domains, such as the sets of rational numbers, real numbers, or complex numbers. Fields used to be called rational domains.

The concept of a field is of use, for example, in defining vectorss and matrices, two structures in linear algebra whose components can be elements of an arbitrary field. Galois theory studies the symmetry of equations by investigating the ways in which fields can be contained in each other. See Field theory (mathematics) for more.

Table of contents
1 Definition
2 Examples of fields
3 Some first theorems
4 Constructing new fields from given ones
5 History
6 Related topics

Definition

A field is a commutative ring (F, +, *) such that 0 does not equal 1 and all elements of F except 0 have a multiplicative inverse.

Spelled out, this means that the following hold:

; Closure of F under + and * : For all a,b belonging to F, both a + b and a * b belong to F (or more formally, + and * are binary operations on F);

; Both + and * are associative : For all a,b,c in F, a + (b + c) = (a + b) + c and a * (b * c) = (a * b) * c.

; Both + and * are commutative : For all a,b belonging to F, a + b = b + a and a * b = b * a.

; The operation * is distributive over the operation + : For all a,b,c, belonging to F, a * (b + c) = (a * b) + (a * c).

; Existence of an additive identity : There exists an element 0 in F, such that for all a belonging to F, a + 0 = a.

; Existence of a multiplicative identity : There exists an element 1 in F different from 0, such that for all a belonging to F, a * 1 = a.

; Existence of additive inverses : For every a belonging to F, there exists an element -a in F, such that a + (-a) = 0.

; Existence of multiplicative inverses : For every a ≠ 0 belonging to F, there exists an element a-1 in F, such that a * a-1 = 1.

The requirement 0 ≠ 1 ensures that the set which only contains a single zero is not a field. Directly from the axioms, one may show that (F, +) and (F - {0}, *) are commutative groups and that therefore (see elementary group theory) the additive inverse -a and the multiplicative inverse a-1 are uniquely determined by a. Furthermore, the multiplicative inverse of a product is equal to the product of the inverses:

(a*b)-1 = a-1 * b-1
provided both a and b are non-zero. Other useful rules include
-a = (-1) * a
and more generally
-(a * b) = (-a) * b = a * (-b)
as well as
a * 0 = 0,
all rules familiar from elementary arithmetic.

Examples of fields

     +  0  1        *  0  1
     0  0  1        0  0  0
     1  1  0        1  0  1

It has important uses in computer science, especially in cryptography and coding theory.

Some first theorems

Constructing new fields from given ones

  1. If a subset E of a field (F,+,*) together with the operations *,+ restricted to E is itself a field, then it is called a subfield of F. Such a subfield has the same 0 and 1 as F.
  2. The polynomial field F(x) is the field of fractions of polynomials in x with coefficients in F.
  3. An algebraic extension of a field F is the smallest field containing F and a root of an irreducible polynomial p(x) in F[x]. Alternatively, it is identical to the factor ring F[x]/<p(x)>, where <p(x)> is the ideal generated by p(x).

History

See
Field theory (mathematics).

Related topics

See Glossary of field theory for more definitions in field theory.