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 subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, Addition is the mathematical process of putting things together Subtraction is one of the four basic Arithmetic operations it is the inverse of Addition, meaning that if we start with any number and add any number and then subtract In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone A number is an Abstract object, tokens of which are Symbols used in Counting and measuring.

All fields are rings, but not conversely. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real Fields differ from rings most importantly in the requirement that division be possible, but also, in modern definitions, by the requirement that the multiplication operation in a field be commutative. In Mathematics, commutativity is the ability to change the order of something without changing the end result Otherwise the structure is a so-called skew field (better known as a division ring), although historically division rings were called fields and fields were commutative fields. In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible

The prototypical example of a field is Q, the field of rational numbers. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions Other important examples include the field of real numbers R, the field of complex numbers C and, for any prime number p, the finite field of integers modulo p, denoted Z/pZ, F_p or GF(p). In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 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, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers For any field K, the set K(X) of rational functions with coefficients in K is also a field. In Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In

The mathematical discipline concerned with the study of fields is called field theory.

A field is a specific type of integral domain, and can be characterized by the following (not necessarily exhaustive) chain of class inclusions:

• integral domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields

Equivalent definitions

Definition 1

A field is a commutative division ring. In Set theory and its applications throughout Mathematics, a subclass is a class contained in some other class in the same way that a Subset In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such In Mathematics, a unique factorization domain (UFD is roughly speaking a Commutative ring in which every element with special exceptions can be uniquely written In Abstract algebra, a principal ideal domain, or PID is an Integral domain in which every ideal is principal i In Abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm applies In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible

Definition 2

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. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property (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. Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity )

Definition 3

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). In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two

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. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, a group G,*> is defined as a set G and a Binary operation on G, called product and denoted Other useful rules include

−a = (−1) * a

and more generally

−(a * b) = (−a) * b = a * (−b)

as well as

a * 0 = 0,

which follows from replacing b or c with 0 in the distributive property

If the requirement of commutativity of the operation * is dropped, one distinguishes the above commutative fields from non-commutative fields. Fields which are not assumed to be commutative are usually called division rings or skew fields. In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible

History

The concept of a field is due to Dedekind, who used the word Körper "body" for this notion. Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important He also was the first to define rings (then called order or order-modul), but the term "a ring" (Zahlring) was invented by Hilbert. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most ^[1]

Examples

• The complex numbers C, under the usual operations of addition and multiplication. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted 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 Q = { a/b | a, b in Z, b ≠ 0 } where Z is the set of integers. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French The field of rational numbers contains no proper subfields.
  • An algebraic number field is a finite field extension of the rational numbers Q, that is, a field containing Q which has finite dimension as a vector space over Q. In Mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) Field extension of the In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Any such field is isomorphic to a subfield of C, and any such isomorphism induces the identity on Q. These fields are very important in number theory. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes
  • The field of algebraic numbers , the algebraic closure of Q. In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or In Mathematics, particularly Abstract algebra, an algebraic closure of a field K is an Algebraic extension of K that is The field of algebraic numbers is an example of an algebraically closed field of characteristic zero; as such it satisfies the same first-order sentences as the field of complex numbers C. In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients In Mathematical logic, a first-order theory is given by a set of axioms in somelanguage
• The real numbers R, under the usual operations of addition and multiplication. In Mathematics, the real numbers may be described informally in several different ways When the real numbers are given the usual ordering, they form a complete ordered field; it is this structure which provides the foundation for most formal treatments of calculus. In Mathematics, an ordered field is a field together with a Total ordering of its elements that agrees in a certain sense with the field operations Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives
  • The real numbers contain several interesting subfields: the real algebraic numbers, the computable numbers. In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or In Mathematics, Theoretical computer science and Mathematical logic, the computable numbers, also known as the recursive numbers or the
• 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, q≠ 1. 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 Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective 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 which has exactly two distinct natural number Divisors 1 (No finite field can exist with any other number of elements. ) This is usually denoted F_q . Finite fields are also called Galois fields. 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 particular, for a given prime number p, the set of integers modulo p is a finite field with p elements: Z/pZ = 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. In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers
  • 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. A Cayley table, after the 19th century British Mathematician Arthur Cayley, describes the structure of a In Logic and/or Mathematics, logical conjunction or and is a two-place Logical operation that results in a value of true if both of Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" Coding theory is one of the most important and direct applications of Information theory.

• The rational numbers can be extended to the fields of p-adic numbers for every prime number p. In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 These fields are important in both number theory and mathematical analysis. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Analysis has its beginnings in the rigorous formulation of Calculus.
• Let E and F be two fields with F a subfield of E. Let x be an element of E not in F. Then there is a smallest subfield of E containing F and x, denoted F(x). We call F(x) a simple extension of F. In Mathematics, more specifically in field theory, a simple extension is a Field extension which is generated by the adjunction of a single element For instance, Q(i) is the subfield of 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 the quotient field of the ring of polynomials F[X]. In Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In In Mathematics, every Integral domain can be embedded in a field; the smallest field which can be used is the field of fractions or field of quotients In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations This is the simplest example of a transcendental extension of F. In Abstract algebra, a Field extension L / K is called algebraic if every element of L is algebraic over K, i
• 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)> , where <p(X)> denotes the ideal generated by p(X), is a field with a subfield isomorphic to F. 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 set In Mathematics, especially in the field of Abstract algebra, a polynomial ring is a ring formed from the set of Polynomials in one or more variables 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. See the primitive element theorem. In Mathematics, more specifically in Field theory, the primitive element theorem provides a characterization of the Finite field extensions which are
• When F is a field, the set F((X)) of formal Laurent series over F is a field. In Mathematics, the Laurent series of a complex function f ( z) is a representation of that function as a Power series which includes terms
• If V is an algebraic variety over F, then the rational functions V → F form a field, the function field of V. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety
• If S is a Riemann surface, then the meromorphic functions S → C form a field. In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional In Complex analysis, a meromorphic function on an open subset D of the Complex plane is a function that is holomorphic
• 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 with respect to U is a field. In the mathematical field of Set theory, an ultrafilter on a set X is a collection of Subsets of X that is a filter, that An ultraproduct is a mathematical construction of which the ultrapower (defined below is a special case
• Hyperreal numbers and superreal numbers extend the real numbers with the addition of infinitesimal and infinite numbers. The superreal numbers are an extension of the Real numbers, similar to the Surreal numbers or Hyperreal numbers but comprising a more inclusive category

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. In Mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers The set of all surreal numbers with birthday smaller than some inaccessible cardinal form a field. In Set theory, an uncountable regular cardinal number is called weakly inaccessible if it is a Weak limit cardinal, and strongly inaccessible
• The nimbers form a Field. In Mathematics, the Proper class of nimbers (occasionally called Grundy numbers) is introduced in Combinatorial game theory, where they are defined The set of nimbers with birthday smaller than , the nimbers with birthday smaller than any infinite cardinal are all examples of fields. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English.

Some first theorems

• The set of non-zero elements of a field F (typically denoted by F^×) is an abelian group under multiplication. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the Every finite subgroup of F^× is cyclic. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an

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

• If there are positive integers n such that 0 = 1 + 1 + . . . + 1 (n repeated terms), then the smallest such n must be a prime number; that is, the characteristic of a field must be either a prime number, or zero. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's

• A commutative ring is a field if and only if it has no ideals except {0} and itself. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring.

• Assuming the axiom of choice, for every field F, there exists a field G which contains F, is unique up to isomorphism inducing the identity on F, is algebraic over F, and is algebraically closed. 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 in one Variable of degree at least 1 with Coefficients G is called the algebraic closure of F. In Mathematics, particularly Abstract algebra, an algebraic closure of a field K is an Algebraic extension of K that is However, in many circumstances in mathematics, it is not appropriate to treat G as being uniquely determined by F, since the isomorphism above is not itself unique. In these cases, one refers to such a G as an algebraic closure of F.

See also

• Glossary of field theory for more definitions in field theory. Field theory is the branch of Mathematics in which fields are studied
• Differential field, a field equipped with a derivation. In Mathematics, differential rings differential fields and differential algebras are rings, fields and algebras equipped with a derivation, In Abstract algebra, a derivation is a function on an algebra which generalizes certain features of the Derivative operator
• Integral domain and its Field of fractions

References

1. ^ J J O'Connor and E F Robertson, The development of Ring Theory, September 2004. In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such In Mathematics, every Integral domain can be embedded in a field; the smallest field which can be used is the field of fractions or field of quotients

External links

• Fields at ProvenMath definition and basic properties.
• Field on PlanetMath