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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


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). 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).
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: 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 ≠ 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.
  • 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.

There are also proper classes with field structure, which are sometimes called Fields, with a capital F: Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, an algebraic number field (or simply number field) is a finite (and therefore algebraic) field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, an ordered field is a field (F,+,*) together with a total order ≤ on F that is compatible with the algebraic operations in the following sense: if a ≤ b then a + c ≤ b + c if 0 ≤ a and 0 ≤ b then 0 ≤ a b It follows from these axioms... Calculus is a central branch of mathematics, developed from algebra and geometry. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... In mathematics, theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers, are the subset of the real numbers consisting of the numbers which can be computed by a finite, terminating algorithm. ... A real number a is first-order definable in the language of set theory, without parameters, if there is a formula φ in the language of set theory, with one free variable, such that a is the unique real number such that φ(a) holds (in the von Neumann universe V). ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ... Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... A Cayley table is a representation of a product defined on a set G. It is a group-theoretic generalization of an addition or a multiplication table. ... Exclusive disjunction (usual symbol xor) is a logical operator that results in true if one of the operands (not both) is true. ... Logical conjunction (usual symbol and) is a logical operator that results in true if both of the operands are true. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... The German Lorenz cipher machine, used in World War II for encryption of very high-level general staff messages. ... Coding theory is a branch of mathematics and computer science dealing with the error-prone process of transmitting data across noisy channels, via clever means, so that a large number of errors that occur can be corrected. ... The title given to this article is incorrect due to technical limitations. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence. ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... A Laurent series is defined with respect to a particular point c and a path of integration γ. The path of integration must lie in an annulus (shown here in red) inside of which f(z) is holomorphic. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. ... In mathematics, especially in order theory, an ultrafilter is a subset of a partially ordered set (a poset) which is maximal among all proper filters. ... An ultraproduct is a mathematical construction, which is used in abstract algebra to construct new fields from given ones, and in model theory, a branch of mathematical logic. ... In mathematics, particularly in non-standard analysis and mathematical logic, hyperreal numbers or nonstandard reals (usually denoted as *R) denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle. ... The superreal numbers compose a more inclusive category than hyperreal number. ...

  • 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 Évariste 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. ...

See also

Wikibooks
Wikibooks Abstract algebra has more about this subject:
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 ≠ 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. ...

External links

  • Fields at ProvenMath definition and basic properties.

  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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