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Encyclopedia > 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. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... Arithmetic is the current mathematics collaboration of the week! Please help improve it to featured article standard. ... A number is an abstract entity that represents a count or measurement. ...

## Contents

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). In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... 0 (zero) is both a number and a numeral. ... Look up one in Wiktionary, the free dictionary. ...

Explicitly, a field is defined by these properties:

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).
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.
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: In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, a group (G,*) is usually defined as: G is a set and * is an associative binary operation on G, obeying the following rules (or axioms): A1. ...

(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. Arithmetic is the current mathematics collaboration of the week! Please help improve it to featured article standard. ...

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. In abstract algebra, a division ring, also called a skew field, is a ring with 0 â‰  1 and such that every non-zero element a has a multiplicative inverse (i. ...

## Examples

• The complex numbers $mathbb C$, 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 $mathbb R$, 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.
• There is (up to isomorphism) exactly one finite field with q elements, for every finite number q which is a power of a prime number. (No field can exist with any other number of elements.) This is usually denoted Fq 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: Z/pZ = Fp = {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 VF form a field, the function field of V.
• If S is a Riemann surface, then the meromorphic functions SC form a field.
• If I is an index set, U is an ultrafilter on I, and Fi is a field for every i in I, the ultraproduct of the Fi (using U) is a field.
• Hyperreal numbers and superreal numbers extend the real numbers with the addition of infinitesimal and infinite numbers.

• 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.

In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar to superreal numbers and hyperreal numbers. ... In mathematics, a cardinal number k > (aleph-null) is called weakly inaccessible, or just inaccessible, if the following two conditions hold. ... In mathematics, the proper class of nimbers is introduced in combinatorial game theory, where they arise as the sizes of nim heaps. ... In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ...

## 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 ZF 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.

In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... In Abstract Algebra, a generator is defined as follows: Let G be a group and , then a is called a generator and G is a cyclic group. ... The word kernel has several meanings in mathematics, some related to each other and some not. ... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ... In abstract algebra, a finite field or Galois field (so named in honor of Ã‰variste Galois) is a field that contains only finitely many elements. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... In abstract algebra, a field extension L /K is called algebraic if every element of L is algebraic over K, i. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero in F. In that case, every such polynomial splits into linear factors. ... In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ... Fields

Image File history File links Wikibooks-logo-en. ... Field theory is a branch of mathematics which studies the properties of fields. ... Field theory is the branch of mathematics in which fields are studied. ... In mathematics, in the area of ring theory, differential rings, differential fields and differential algebras are rings, fields and algebras equipped with a derivation. ... In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 â‰  1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the field of fractions of the integral domain. ... Results from FactBites:

 Field (mathematics) - Wikipedia, the free encyclopedia (1374 words) 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. 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. Differential field, a field equipped with a derivation.
 Field theory (mathematics) - Wikipedia, the free encyclopedia (615 words) Field theory is a branch of mathematics which studies the properties of fields. 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. For instance, the field of algebraic numbers is the algebraic closure of the field of rational numbers and the field of complex numbers is the algebraic closure of the field of real numbers.
More results at FactBites »

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