Field (mathematics)

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.

When abstract algebra was first being developed, the definition of field usually did not include commutativity, and a modern field would have been called either a commutative field or a rational domain. In contemporary usage, a field is always commutative. An old-style field, that is, an object which satisfies all the properties of a field except for commutativity, is now called a division ring or sometimes a skew field.

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 symmetry of equations by investigating the ways in which fields can be contained in each other. See field theory for more.

Contents

1 Definition

2 Examples of fields

3 Some first theorems

4 Constructing new fields from given ones

5 See also

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 (abelian 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 = b⁻¹ * a⁻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.

If the requirement of commutativity of the operation * is dropped, one distinguishes the above commutative fields from non-commutative fields, usually called division rings or skew fields).

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

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

More generally: if q > 1 is a power of a prime number, then there exists (up to isomorphism) exactly one finite field with q elements. No other finite fields exist. 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. Such fields are often called Galois fields.

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.

The rational numbers can be extended to the fields of p-adic numbers for every prime number p.

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.

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.

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).

When F is a field, the set F((X)) of formal Laurent series over F is a field.

If V is an algebraic variety over F, then the rational functions V → F form a field, the function field of V.

If S is a Riemann surface, then the meromorphic functions S → C form a field.

If I is an index set, U is an ultrafilter on I, and F_i is a field for every i in I, the ultraproduct of the F_i (using U) is a field.

Hyperreal numbers and superreal numbers extend the real numbers with the addition of infinitesimal and infinite numbers.

There are also proper classes with field structure, which are some times called Fields.

The surreal numbers form a Field containing the reals, and would be a field 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. 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.

Some first theorems

The set of non-zero elements of a field F (typically denoted by F^×) is an abelian group under multiplication. Every finite subgroup of F^× is cyclic.

The characteristic of any field is zero or a prime number. (The characteristic is defined as the smallest positive integer n such that n·1 = 0, or zero if no such n exists; here n·1 stands for n summands 1 + 1 + 1 + ... + 1.)

The number of elements in finite fields is a prime power.

As a ring, a field has no ideals except {0} and itself.

For every field F, there exists a (up to isomorphism) unique field G which contains F, is algebraic over F, and is algebraically closed. G is called the algebraic closure of F.

Constructing new fields from given ones

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.
The polynomial field F(x) is the field of fractions of polynomials in x with coefficients in F.
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).