Field

A field, in abstract algebra, is an algebraic system of elements in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed without leaving the system and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. Fields are important objects of study in abstract algebra since they provide the proper generalization of number domains, such as the sets of rational numbers or real numbers. A formally correct treatment of the concept follows.

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) and (b + c) * a = (b * a) + (c * a).

Existence of an additive identity

There exists an element 0 in F, such that for all a belonging to F, a + 0 = a and 0 + a= 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 and 1 * a = a.

Existence of an additive inverse

For all a belonging to F, there exists an element -a in F, such that a + (-a) = 0 and (-a) + a = 0.

Existence of a multiplicative inverse

For all a <> 0 belonging to F, there exists an element a^-1 in F, such that a * a^-1 = 1 and a^-1 * a = 1.

Directly from these axioms, one may show that (F, +) and (F - {0}, *) are commutative groups and that therefore the additive inverse -a and the multiplicative invers a^-1 are uniquely determined by a. Furthermore, the inverse of a product is equal to the product of the inverses (see elementary group theory).

Examples of Fields.

• The rational numbers Q = { a/b | a, b in Z, b <> 0 } where Z is the set of integers.

• The real numbers R .

• The complex numbers C.

• The smallest field has only two elements: 0 and 1. It is sometimes denoted by F₂ or Z₂ and can be defined by the two tables

      +  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]?.

• More generally: there exists exactly one finite field (up to isomorphism) of order q if q > 1 is an integer power of a prime number p. For instance, for a prime number p, the set of integers modulo p is a finite field with p elements: this is often written as Z_p = {0,1,...,p-1} where the operations are defined by performing the operation in Z, dividing by p and taking the remainder, see modular arithmetic.

• The real numbers contain several interesting fields: the real algebraic numbers, the computable numbers, and the definable numbers.

• The complex numbers contain the field of algebraic numbers, the [algebraic closure]? of Q.

• For a given field F, the set F(X) of rational functions in the variable X with coefficients in F is a field; this is defined as the set of quotients of polynomials with coefficients in F.

• Let E and F be two fields with E a subfield of F (i.e., a subset of F containing 0 and 1, closed under the operations + and * of F and with its own operations defined by restriction). Let x be an element of F not in E. Then E(x) is defined to be the smallest subfield of F containing E and x. For instance, Q(i) is the subfield of the complex numbers C consisting of all numbers of the form a+bi where both a and b are rational numbers.

• If F is a field, and p(X) is an [irreducible polynomial]? in the polynomial ring F[X], then the quotient F[X]/(p(X)) is a field with a subfield isomorphic to F. For instance, R[X]/(X²+1) is a field (in fact, it is isomorphic to the field of complex numbers).

• The hyperreal numbers form a field containing the reals, plus infinitesimal and infinite numbers.

• The surreal numbers form a field containing the reals, except for the fact that they are a proper class, not a set. The set of all surreal numbers with birthday smaller than some inaccessible cardinal number form a field.

• The nimbers? form a field, again except for the fact that they are a proper class. The set of nimbers with birthday smaller than 2^(2^n), the nimbers with birthday smaller than any infinite cardinal are all examples of fields.

The concept of a field is of use, for example, in defining vectors and matrices, two structures in Linear Algebra, whose components can be elements of an arbitrary field. [Galois theory]? studies the ways in which fields can be contained in each other.

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