Field (mathematics)

Table of Contents

• 1 Introduction
• 2 Definition
• 3 Examples of fields
• 4 Some first theorems
• 5 See also

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 number s. 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 fieldor a rational domain. In contemporary usage, a field is always commutative. A structure which satisfies all the properties of a field except for commutativity, is today called a division ring or sometimes a skew field, but also non-commutative fieldis still widely used. Other languages have retained the old usage: for example, in Italian and French , division rings are called corpoand corps, both literally meaning 'body'. Instead, in German and Spanish , Körper(whence the blackboard bold Kused to denote a field) and cuerpomean 'field'. Notice that 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 vector s 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.

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

Spelled out, this means that the following hold:

all rules familiar from elementary arithmetic .

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

• The complex numbers C, under the usual operations of addition and multiplication. The field of complex numbers contains the following subfields(a subfield of a field Fis a set containing 0 and 1, closed under the operations + and * of Fand with its own operations defined by restriction):
• The rational numbers Q= { a/ b| a, bin Z, b≠ 0 } where Zis the set of integers . The rational number field contains no proper subfields.
• An algebraic number field is a finite field extension of the rational number s Q, that is, a field containing Qwhich has finite dimension as a vector space over Q. Such fields are very important in number theory .
• The field of algebraic numbers , the algebraic closure of Q.
• The real numbers R, under the usual operations of addition and multiplication. When the real numbers are given the usual ordering, they form a complete ordered fieldwhich 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 number s, the computable number s, and the definable number s.
• If q> 1 is a power of a prime number , then there exists ( up to isomorphism ) exactly one finite field with qelements, 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 pis a finite field with pelements: F_p= {0, 1, ..., p − 1} where the operations are defined by performing the operation in Z, dividing by pand 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

• 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 Eand Fbe two fields with Ea subfield of F. Let xbe an element of Fnot in E. Then E( x) is defined to be the smallest subfield of Fcontaining Eand x. We call E( x) a simple extensionof E. For instance, Q( i) is the number field of complex numbers Cconsisting of all numbers of the form a+ biwhere both aand bare 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 function s in the variable Xwith coefficients in Fis 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 Fis 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 Fis isomorphic to a field of this form.
• When Fis a field, the set F(( X)) of formal Laurent series over Fis a field.
• If Vis an algebraic variety over F, then the rational functions V→ Fform a field, the function fieldof V.
• If Sis a Riemann surface , then the meromorphic function s S→ Cform a field.
• If Iis an index set, Uis an ultrafilter on I, and F_iis a field for every iin I, the ultraproduct of the F_i(using U) is a field.
• Hyperreal numbers and superreal number s 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.

• 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 nsuch that n·1 = 0, or zero if no such nexists; here n·1 stands for nsummands 1 + 1 + 1 + ... + 1. An equivalent definition is the following: the characteristic of a field Fis the unique non-negative generator of the kernel of the unique ring homomorphism Z→ Fwhich sends 1 |-> 1.)

• The number of elements of any finite field is a prime power.
• As a ring, a field has no ideal s 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 . Gis called the algebraic closure of F.

• field theory for some history and other information.
• Glossary of field theory for more definitions in field theory.