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 same rules hold which are familiar from the arithmetic of ordinary numbers.

Introduction

Fields are important objects of study in algebra since they provide a useful generalization of many number systems, such as the rational numbers, real numbers, and complex numbers. In particular, the usual rules of associativity, commutativity and distributivity hold. Fields also appear in many other areas of mathematics; see the examples below.

When abstract algebra was first being developed, the definition of a field usually did not include commutativity of multiplication, and what we today call a field would have been called either a commutative field or a rational domain. In contemporary usage, a field is always commutative. A structure which satisfies all the properties of a field except possibly for commutativity, is today called a division ring or sometimes a skew field, but also non-commutative field is still widely used.

The German word for field is Körper (literally, body); hence the use of the blackboard bold $\mathbb {K}$ to denote a field. In Spanish, cuerpo (with the same literal meaning) is used. However, other languages have retained the old usage. In Italian and French, division rings are called corpo and corps, both literally meaning 'body'. The French language has no single word for field; they are simply called corps commutatif. Italian for field is campo, with the same literal meaning as English.

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 information.

History

In 1910 Ernst Steinitz gave the first axiomatic definition of a field in his paper Algebraische Theorie der Körper (German:algebraic theory of fields).

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. (Note that 0 and 1 here stand for the identity elements for the + and * operations respectively, which may differ from the familiar real numbers 0 and 1).

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 element 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 complex numbers $\mathbb {C}$ $\mathbb {C}$ , under the usual operations of addition and multiplication. The field of complex numbers contains the following subfields (a subfield of a field F is a set containing 0 and 1, closed under the operations + and * of F and with its own operations defined by restriction):
- The rational numbers $\mathbb {Q}$ = { a/b | a, b in $\mathbb {Z}$ , b ≠ 0 } where $\mathbb {Z}$ is the set of integers. The rational number field contains no proper subfields.
- An algebraic number field is a finite field extension of the rational numbers $\mathbb {Q}$ , that is, a field containing $\mathbb {Q}$ which has finite dimension as a vector space over $\mathbb {Q}$ . Such fields are very important in number theory.
- The field of algebraic numbers, the algebraic closure of $\mathbb {Q}$ .
- The real numbers $\mathbb {R}$ $\mathbb {R}$ , under the usual operations of addition and multiplication. When the real numbers are given the usual ordering, they form a complete ordered field which is categorical — it is this structure that provides the foundation for most formal treatments of calculus.
  - The real numbers contain several interesting subfields: the real algebraic numbers, the computable numbers, and the definable numbers.
If q > 1 is a power of a prime number, then there exists (up to isomorphism) exactly one finite field with q elements, usually denoted F_q, Z/qZ, or GF(q). Every other finite field is isomorphic to one of these fields. Such fields are often called a Galois field, whence the notation GF(q).
- In particular, for a given prime number p, the set of integers modulo p is a finite field with p elements: F_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.
  - Taking p = 2, we obtain the smallest field, F₂, which has only two elements: 0 and 1. It can be defined by the two Cayley 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.

The rational numbers can be extended to the fields of p-adic numbers for every prime number p. These fields are very important in both number theory and mathematical analysis.
Let E and F be two fields with F a subfield of E. Let x be an element of E not in F. Then F(x) is defined to be the smallest subfield of E containing F and x. We call F(x) a simple extension of F. For instance, Q(i) is the number field of complex numbers C consisting of all numbers of the form a + bi where both a and b are rational numbers. In fact, it can be shown that every number field is a simple extension 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. This is the simplest example of a transcendental extension.
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). It can be shown that every simple algebraic extension of F is isomorphic to a field of this form.
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 sometimes called Fields, with a capital F:

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 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 follows: 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. An equivalent definition is the following: the characteristic of a field F is the unique non-negative generator of the kernel of the unique ring homomorphism Z → F which sends 1 |-> 1.)

The number of elements of any finite field is a prime power.

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

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

Introduction

History

Definition

Examples of fields

Some first theorems

See also