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| {{about|fields in algebra|fields in geometry|Vector field|1|Field (disambiguation)}}
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| In [[abstract algebra]], a '''field''' is a [[zero ring|nonzero]] [[commutative ring]] that contains a [[multiplicative inverse]] for every [[nonzero element in ring|nonzero]] element, or equivalently a [[ring (mathematics)|ring]] whose nonzero elements form an [[abelian group]] under multiplication. As such it is an [[algebraic structure]] with notions of [[addition]], [[subtraction]], [[multiplication]], and [[division (mathematics)|division]] satisfying the appropriate abelian group equations and [[distributive law]]. The most commonly used fields are the field of [[real number]]s, the field of [[complex number]]s, and the field of [[rational number]]s, but there are also [[finite field]]s, fields of [[function (mathematics)|functions]], [[algebraic number field]]s, [[p-adic number|''p''-adic fields]], and so forth.
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| Any field may be used as the [[scalar (mathematics)|scalars]] for a [[vector space]], which is the standard general context for [[linear algebra]]. The theory of [[field extension]]s (including [[Galois theory]]) involves the [[root of a function|roots]] of [[polynomial]]s with [[coefficient]]s in a field; among other results, this theory leads to impossibility proofs for the classical problems of [[angle trisection]] and [[squaring the circle]] with a [[compass and straightedge]], as well as a proof of the [[Abel–Ruffini theorem]] on the algebraic insolubility of [[quintic equation]]s. In modern mathematics, the theory of fields (or '''[[field theory (mathematics)|field theory]]''') plays an essential role in [[number theory]] and [[algebraic geometry]].
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| As an algebraic structure, every field is a [[ring (mathematics)|ring]], but not every ring is a field. The most important difference is that fields allow for division (though not [[division by zero]]), while a ring need not possess [[multiplicative inverse]]s; for example the [[Ring of integers|integers]] form a ring, but 2''x'' = 1 has no solution in integers. Also, the multiplication operation in a field is required to be [[commutative]]. A ring in which division is possible but commutativity is not assumed (such as the [[quaternion]]s) is called a ''[[division ring]]'' or ''skew field''. (Historically, division rings were sometimes referred to as fields, while fields were called ''commutative fields''.)
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| As a ring, a field may be classified as a specific type of [[integral domain]], and can be characterized by the following (not exhaustive) chain of [[subclass (set theory)|class inclusions]]:
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| : '''[[Commutative ring]]s''' ⊃ '''[[integral domain]]s''' ⊃ '''[[integrally closed domain]]s''' ⊃ '''[[unique factorization domain]]s''' ⊃ '''[[principal ideal domain]]s''' ⊃ '''[[Euclidean domain]]s''' ⊃ '''fields''' ⊃ '''[[finite field]]s'''.
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| == Definition and illustration ==
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| {{Algebraic structures |Ring}}
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| Intuitively, a field is a set ''F'' that is a commutative group with respect to two compatible operations, addition and multiplication, with "compatible" being formalized by ''distributivity,'' and the caveat that the additive identity (0) has no multiplicative inverse (one cannot [[Division by zero|divide by 0]]).
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| The most common way to formalize this is by defining a ''field'' as a [[set (mathematics)|set]] together with two [[binary operation|operations]], usually called ''addition'' and ''multiplication'', and denoted by + and ·, respectively, such that the following axioms hold; ''subtraction'' and ''division'' are defined implicitly in terms of the inverse operations of addition and multiplication:<ref group="note">That is, the axiom for addition only assumes a binary operation <math>\scriptstyle +\colon\, F \,\times\, F \;\to\; F,\,</math> <math>\scriptstyle a,\, b \;\mapsto\; a \,+\, b.</math>The axiom of inverse allows one to define a unary operation <math>\scriptstyle -\colon\, F \;\to\; F</math> <math>\scriptstyle a \;\mapsto\; -a</math> that sends an element to its negative (its additive inverse); this is not taken as given, but is implicitly defined in terms of addition as "<math>\scriptstyle -a</math> is the unique ''b'' such that <math>\scriptstyle a \,+\, b \;=\; 0</math>", "implicitly" because it is defined in terms of solving an equation—and one then defines the binary operation of subtraction, also denoted by "−", as <math>\scriptstyle -\colon F \,\times\, F \;\to\; F,\,</math> <math>\scriptstyle a,\, b \;\mapsto\; a \,-\, b \;:=\; a \,+\, (-b)</math> in terms of addition and additive inverse.
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| In the same way, one defines the binary operation of division ÷ in terms of the assumed binary operation of multiplication and the implicitly defined operation of "reciprocal" (multiplicative inverse).</ref>
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| ;''Closure'' of ''F'' under addition and multiplication
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| :For all ''a'', ''b'' in ''F'', both ''a'' + ''b'' and ''a'' · ''b'' are in ''F'' (or more formally, + and · are [[binary operations]] on ''F'').
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| ;''[[Associativity]]'' of addition and multiplication
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| :For all ''a'', ''b'', and ''c'' in ''F'', the following equalities hold: {{nowrap begin}}''a'' + (''b'' + ''c'') = (''a'' + ''b'') + ''c''{{nowrap end}} and {{nowrap begin}}''a'' · (''b'' · ''c'') = (''a'' · ''b'') · ''c''{{nowrap end}}.
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| ;''[[Commutativity]]'' of addition and multiplication
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| :For all ''a'' and ''b'' in ''F'', the following equalities hold: {{nowrap begin}}''a'' + ''b'' = ''b'' + ''a''{{nowrap end}} and {{nowrap begin}}''a'' · ''b'' = ''b'' · ''a''{{nowrap end}}.
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| ; Existence of additive and multiplicative ''[[identity element]]s''
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| :There exists an element of ''F'', called the ''additive identity'' element and denoted by 0, such that for all ''a'' in ''F'', {{nowrap begin}}''a'' + 0 = ''a''{{nowrap end}}. Likewise, there is an element, called the ''multiplicative identity'' element and denoted by 1, such that for all ''a'' in ''F'', {{nowrap begin}}''a'' · 1 = ''a''{{nowrap end}}. To exclude the [[trivial ring]], the additive identity and the multiplicative identity are required to be distinct.
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| ;Existence of ''[[additive inverse]]s'' and ''[[multiplicative inverse]]s''
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| :For every ''a'' in ''F'', there exists an element −''a'' in ''F'', such that {{nowrap begin}}''a'' + (−''a'') = 0{{nowrap end}}. Similarly, for any ''a'' in ''F'' other than 0, there exists an element ''a''<sup>−1</sup> in ''F'', such that {{nowrap begin}}''a'' · ''a''<sup>−1</sup> = 1{{nowrap end}}. (The elements ''a'' + (−''b'') and ''a'' · ''b''<sup>−1</sup> are also denoted ''a'' − ''b'' and ''a''/''b'', respectively.) In other words, ''subtraction'' and ''division'' operations exist.
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| ;''[[Distributivity]]'' of multiplication over addition
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| :For all ''a'', ''b'' and ''c'' in ''F'', the following equality holds: {{nowrap begin}}''a'' · (''b'' + ''c'') = (''a'' · ''b'') + (''a'' · ''c''){{nowrap end}}.
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| A field is therefore an [[algebraic structure]] 〈''F'', +, ·, −, <sup>−1</sup>, 0, 1〉; of type 〈2, 2, 1, 1, 0, 0〉, consisting of two [[abelian group]]s:
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| * ''F'' under +, −, and 0;
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| * ''F'' \ {0} under ·, <sup>−1</sup>, and 1, with 0 ≠ 1,
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| with · distributing over +.<ref>Wallace, D A R (1998) ''Groups, Rings, and Fields'', SUMS. Springer-Verlag: 151, Th. 2.</ref>
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| ===First example: rational numbers===
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| A simple example of a field is the field of rational numbers, consisting of numbers which can be written as [[fraction (mathematics)|fractions]]
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| ''a''/''b'', where ''a'' and ''b'' are [[integer]]s, and ''b'' ≠ 0. The additive inverse of such a fraction is simply −''a''/''b'', and the multiplicative inverse (provided that ''a'' ≠ 0) is ''b''/''a''. To see the latter, note that
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| :<math>\frac{b}{a} \cdot \frac{a}{b} = \frac{ba}{ab} = 1.</math>
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| The abstractly required field axioms reduce to standard properties of rational numbers, such as the law of [[distributivity]]
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| :<math>\frac{a}{b} \cdot \left(\frac{c}{d} + \frac{e}{f}\right)</math>
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| :<math>= \frac{a}{b} \cdot \left(\frac{c}{d} \cdot \frac{f}{f} + \frac{e}{f} \cdot \frac{d}{d}\right) </math>
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| :<math>= \frac{a}{b} \cdot \left(\frac{cf}{df} + \frac{ed}{fd}\right) = \frac{a}{b} \cdot \frac{cf + ed}{df}</math>
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| :<math>= \frac{a(cf + ed)}{bdf} = \frac{acf}{bdf} + \frac{aed}{bdf} = \frac{ac}{bd} + \frac{ae}{bf}</math>
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| :<math>= \frac{a}{b} \cdot \frac{c}{d} + \frac{a}{b}\cdot \frac{e}{f}\text{,}</math>
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| or the law of [[commutativity]] and law of [[associativity]].
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| ===Second example: a field with four elements===
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| {| class="wikitable" style="float:right; text-align:center;"
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| ! style="width:20%;"| · !! style="width:20%;"| O !! style="width:20%;"| I !! style="width:20%;"| A !! style="width:20%;"| B
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| | style="background:#fdd; border-left:solid black 2px; border-top:solid black 2px;"| O
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| |}
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| In addition to familiar number systems such as the rationals, there are other, less immediate examples of fields. The following example is a field consisting of four elements called O, I, A and B. The notation is chosen such that O plays the role of the additive identity element (denoted 0 in the axioms), and I is the multiplicative identity (denoted 1 above). One can check that all field axioms are satisfied. For example:
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| :A · (B + A) = A · I = A, which equals A · B + A · A = I + B = A, as required by the distributivity.
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| The above field is called a [[finite field]] with four elements, and can be denoted '''F'''<sub>4</sub>. Field theory is concerned with understanding the reasons for the existence of this field, defined in a fairly ad-hoc manner, and describing its inner structure. For example, from a glance at the multiplication table, it can be seen that any non-zero element (i.e., I, A, and B) is a power of A: {{nowrap begin}}A = A<sup>1</sup>{{nowrap end}}, {{nowrap begin}}B = A<sup>2</sup> = A · A,{{nowrap end}} and finally {{nowrap begin}}I = A<sup>3</sup> = A · A · A.{{nowrap end}} This is not a coincidence, but rather one of the starting points of a deeper understanding of (finite) fields.
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| ===Alternative axiomatizations===
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| As with other algebraic structures, there exist alternative axiomatizations. Because of the relations between the operations, one can alternatively axiomatize a field by explicitly assuming that there are four binary operations (add, subtract, multiply, divide) with axioms relating these, or (by [[functional decomposition]]) in terms of two binary operations (add and multiply) and two unary operations (additive inverse and multiplicative inverse), or other variants.
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| The usual axiomatization in terms of the two operations of addition and multiplication is brief and allows the other operations to be defined in terms of these basic ones, but in other contexts, such as [[topology]] and [[category theory]], it is important to include all operations as explicitly given, rather than implicitly defined (compare [[topological group]]). This is because without further assumptions, the implicitly defined inverses may not be [[continuous function|continuous]] (in topology), or may not be able to be defined (in category theory). Defining an inverse requires that one is working with a set, not a more general object.
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| For a very economical axiomatization of the field of [[real number]]s, whose primitives are merely a set '''R''' with 1∈'''R''', addition, and a [[binary relation]], "<". See [[Tarski's axiomatization of the reals]].
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| ==Related algebraic structures==
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| {{Ring-like structures}}
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| The axioms imposed above resemble the ones familiar from other algebraic structures. For example, the existence of the binary operation "·", together with its commutativity, associativity, (multiplicative) identity element and inverses are precisely the axioms for an [[abelian group]]. In other words, for any field, the subset of nonzero elements ''F'' \ {0}, also often denoted ''F''<sup>×</sup>, is an abelian group (''F''<sup>×</sup>, ·) usually called [[multiplicative group]] of the field. Likewise {{nowrap|(''F'', +)}} is an abelian group. The structure of a field is hence the same as specifying such two group structures (on the same set), obeying the distributivity.
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| Important other algebraic structures such as [[ring (mathematics)|rings]] arise when requiring only part of the above axioms. For example, if the requirement of commutativity of the multiplication operation · is dropped, one gets structures usually called [[division ring]]s or ''skew fields''.
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| ===Remarks===
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| By [[elementary group theory]], applied to the abelian groups (''F''<sup>×</sup>, ·), and {{nowrap|(''F'', +)}}, the additive inverse −''a'' and the multiplicative inverse ''a''<sup>−1</sup> are uniquely determined by ''a''.
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| Similar direct consequences from the field axioms include
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| :−(''a · b'') = (−''a'') · b = ''a'' · (−''b''), in particular −''a'' = (−1) · ''a''
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| as well as
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| :''a'' · 0 = 0.
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| Both can be shown by replacing ''b'' or ''c'' with 0 in the distributive property
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| ==History==
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| The concept of ''field'' was used implicitly by [[Niels Henrik Abel]] and [[Évariste Galois]] in their work on the solvability of polynomial equations with rational coefficients of degree five or higher.
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| In 1857, [[Karl von Staudt]] published his [[Karl von Staudt#Algebra of throws|Algebra of Throws]] which provided a geometric model satisfying the axioms of a field. This construction has been frequently recalled as a contribution to the [[foundations of mathematics]].
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| In 1871, [[Richard Dedekind]] introduced, for a set of real or complex numbers which is closed under the four arithmetic operations, the [[German (language)|German]] word ''Körper'', which means "body" or "corpus" (to suggest an organically closed entity), hence the common use of the letter ''K'' to denote a field. He also defined rings (then called ''order'' or ''order-modul''), but the term ''"a ring"'' (''Zahlring'') was invented by [[David Hilbert|Hilbert]].<ref>J J O'Connor and E F Robertson, [http://www-history.mcs.st-andrews.ac.uk/HistTopics/Ring_theory.html ''The development of Ring Theory''], September 2004.</ref> In 1893, [[E. H. Moore|Eliakim Hastings Moore]] called the concept "field" in English.<ref>[http://jeff560.tripod.com/f.html ''Earliest Known Uses of Some of the Words of Mathematics (F)'']</ref>
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| In 1881, [[Leopold Kronecker]] defined what he called a "domain of rationality", which is indeed a field of polynomials in modern terms. In 1893, [[Heinrich M. Weber]] gave the first clear definition of an abstract field.<ref>{{Citation | last1=Fricke | first1=Robert | last2=Weber | first2=Heinrich Martin | author2-link=Heinrich Martin Weber | title=Lehrbuch der Algebra | url=http://resolver.sub.uni-goettingen.de/purl?PPN234788267 | publisher=Vieweg | year=1924}}</ref> In 1910, [[Ernst Steinitz]] published the very influential paper ''Algebraische Theorie der Körper'' ({{lang-en|Algebraic Theory of Fields}}).<ref>{{Citation | last1=Steinitz | first1=Ernst | author1-link=Ernst Steinitz | title=Algebraische Theorie der Körper | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002167042 | year=1910 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=137 | pages=167–309}}</ref> In this paper he axiomatically studies the properties of fields and defines many important field theoretic concepts like [[prime field]], [[perfect field]] and the [[transcendence degree]] of a [[field extension]].
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| [[Emil Artin]] developed the relationship between groups and fields in great detail from 1928 through 1942.
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| <!--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 ''division algebra'' or sometimes a ''skew field''. Also ''non-commutative field'' is still widely used. In the [[French (language)|French language]], fields are called ''corps'' (literally, ''body''), skew fields are called ''corps gauche'' or ''anneau à divisions'' or also ''algèbre à divisions''. -->
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| ==Examples==
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| ===Rationals and algebraic numbers===
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| The field of [[rational number]]s '''Q''' has been introduced above. A related class of fields very important in [[number theory]] are [[algebraic number field]]s. We will first give an example, namely the field '''Q'''(ζ) consisting of numbers of the form
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| :''a'' + ''b''ζ
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| with ''a'', ''b'' ∈ '''Q''', where ζ is a primitive third [[root of unity]], i.e., a complex number satisfying {{nowrap begin}}ζ<sup>3</sup> = 1{{nowrap end}}, {{nowrap|ζ ≠ 1}}. This field extension can be used to prove a special case of [[Fermat's last theorem]], which asserts the non-existence of rational nonzero solutions to the equation
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| :''x''<sup>3</sup> + ''y''<sup>3</sup> = ''z''<sup>3</sup>.
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| In the language of field extensions detailed below, '''Q'''(ζ) is a field extension of degree 2. Algebraic number fields are by definition finite field extensions of '''Q''', that is, fields containing '''Q''' having finite dimension as a '''Q'''-[[vector space]]. <!--Any such field is isomorphic to a subfield of '''C''', and any such isomorphism induces the identity on '''Q'''.-->
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| ===Reals, complex numbers, and ''p''-adic numbers===
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| Take the [[real number]]s '''R''', under the usual operations of addition and multiplication. 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]].
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| The [[complex number]]s '''C''' consist of expressions
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| :''a'' + ''b''i
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| where i is the [[imaginary unit]], i.e., a (non-real) number satisfying i<sup>2</sup> = −1.
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| Addition and multiplication of real numbers are defined in such a way that all field axioms hold for '''C'''. For example, the distributive law enforces
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| :(''a'' + ''b''i)·(''c'' + ''d''i) = ''ac'' + ''bc''i + ''ad''i + ''bd''i<sup>2</sup>, which equals ''ac''−''bd'' + (''bc'' + ''ad'')i.
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| The real numbers can be constructed by [[completion (metric space)|completing]] the rational numbers, i.e., filling the "gaps": for example √<span style="text-decoration:overlin">2</span> is such a gap. By a formally very similar procedure, another important class of fields, the field of [[p-adic number|''p''-adic numbers]] '''Q'''<sub>''p''</sub> is built. It is used in number theory and [[p-adic analysis|''p''-adic analysis]].
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| [[Hyperreal numbers]] and [[superreal number]]s extend the real numbers with the addition of infinitesimal and infinite numbers.
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| ===Constructible numbers===
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| [[File:Multiplication intercept theorem.svg|right|thumb|200px|Given 0, 1, ''r''<sub>1</sub> and ''r''<sub>2</sub>, the construction yields ''r''<sub>1</sub>·''r''<sub>2</sub>]]
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| In antiquity, several geometric problems concerned the (in)feasibility of constructing certain numbers with [[compass and straightedge]]. For example it was unknown to the Greeks that it is in general impossible to trisect a given angle. Using the field notion and field theory allows these problems to be settled. To do so, the field of [[constructible numbers]] is considered. It contains, on the plane, the points 0 and 1, and all complex numbers that can be constructed from these two by a finite number of construction steps using only [[compass]] and [[straightedge]]. This set, endowed with the usual addition and multiplication of complex numbers does form a field. For example, multiplying two (real) numbers ''r''<sub>1</sub> and ''r''<sub>2</sub> that have already been constructed can be done using construction at the right, based on the [[intercept theorem]]. This way, the obtained field ''F'' contains all rational numbers, but is bigger than '''Q''', because for any ''f'' ∈ ''F'', the [[square root]] of ''f'' is also a constructible number.
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| A closely related concept is that of a [[Euclidean field]], namely an [[ordered field]] whose positive elements are closed under square root. The real constructible numbers form the least Euclidean field, and the Euclidean fields are precisely the ordered extensions thereof.
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| ===Finite fields===
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| {{main|Finite field}}
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| ''[[Finite field]]s'' (also called ''Galois fields'') are fields with finitely many elements. The above introductory example '''F'''<sub>4</sub> is a field with four elements. '''F'''<sub>2</sub> consists of two elements, 0 and 1. This is the smallest field, because by definition a field has at least two distinct elements 1 ≠ 0. Interpreting the addition and multiplication in this latter field as [[XOR]] and [[Logical AND|AND]] operations, this field finds applications in [[computer science]], especially in [[cryptography]] and [[coding theory]].
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| In a finite field there is necessarily an integer ''n'' such that {{nowrap|1 + 1 + ··· + 1}} (''n'' repeated terms) equals 0. It can be shown that the smallest such ''n'' must be a [[prime number]], called the ''[[characteristic (algebra)|characteristic]]'' of the field. If a (necessarily infinite) field has the property that {{nowrap|1 + 1 + ··· + 1}} is never zero, for any number of summands, such as in '''Q''', for example, the characteristic is said to be zero.
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| A basic class of finite fields are the fields '''F'''<sub>''p''</sub> with ''p'' elements (''p'' a prime number):
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| :{{nowrap begin}}'''F'''<sub>''p''</sub> = '''Z'''/''p'''''Z''' = {0, 1, ..., ''p'' − 1},{{nowrap end}}
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| where the operations are defined by performing the operation in the set of [[integer]]s '''Z''', dividing by ''p'' and taking the remainder; see [[modular arithmetic]]. A field '''K''' of characteristic ''p'' necessarily contains '''F'''<sub>''p''</sub>,<ref>Jacobson (2009), p. 213</ref> and therefore may be viewed as a [[vector space]] over '''F'''<sub>''p''</sub>, of finite [[Dimension (vector space)|dimension]] if '''K''' is finite. Thus a finite field '''K''' has prime power order, i.e., '''K''' has ''q'' = ''p''<sup>''n''</sup> elements (where {{nowrap begin}}''n'' > 0{{nowrap end}} is the number of elements in a basis of '''K''' over '''F'''<sub>''p''</sub>). By developing more field theory, in particular the notion of the [[splitting field]] of a polynomial ''f'' over a field '''K''', which is the smallest field containing '''K''' and all roots of ''f'', one can show that two finite fields with the same number of elements are isomorphic, i.e., there is a one-to-one mapping of one field onto the other that preserves multiplication and addition. Thus we may speak of ''the'' finite field with ''q'' elements, usually denoted by '''F'''<sub>''q''</sub> or GF(''q'').
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| ===Archimedean fields===
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| {{main|Archimedean field}}
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| An Archimedean field is an [[ordered field]] such that for each element there exists a finite expression {{nowrap|1 + 1 + ··· + 1}} whose value is greater than that element, that is, no infinite elements. Equivalently, the field contains no [[infinitesimals]]; or, the field is isomorphic to a [[subfield]] of the reals. A necessary condition for an ordered field to be [[complete field|complete]] is that it be Archimedean, since in any non-Archimedean field there is neither a greatest infinitesimal nor a least positive rational, whence the sequence 1/2, 1/3, 1/4, …, every element of which is greater than every infinitesimal, has no limit. (And since every proper subfield of the reals also contains such gaps, up to isomorphism the reals form the unique complete ordered field.)
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| ===Field of functions===
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| Given a [[geometry|geometric]] object ''X'', one can consider [[function (mathematics)|functions]] on such objects. Adding and multiplying them pointwise, i.e., {{nowrap begin}}(''f''·''g'')(''x'') = ''f''(''x'') · ''g''(''x''){{nowrap end}} this leads to a field. However, due to the presence of possible zeros, i.e., points ''x'' ∈ ''X'' where ''f''(''x'') = 0, one has to take [[Pole (complex analysis)|poles]] into account, i.e., formally allowing ''f''(''x'') = ∞.
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| If ''X'' is an [[algebraic variety]] over ''F'', then the [[rational function]]s ''X'' → ''F'', i.e., functions defined [[Zariski topology|almost everywhere]], form a field, the [[Function field of an algebraic variety|function field]] of ''X''. Likewise, if ''X'' is a [[Riemann surface]], then the [[meromorphic function]]s ''S'' → '''C''' form a field. Under certain circumstances, namely when ''S'' is [[compact topological space|compact]], ''S'' can be reconstructed from this field.
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| ===Local and global fields===
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| Another important distinction in the realm of fields, especially with regard to number theory, are [[local field]]s and [[global field]]s. Local fields are completions of global fields at a given place. For example, '''Q''' is a global field, and the attached local fields are '''Q'''<sub>''p''</sub> and '''R''' ([[Ostrowski's theorem]]). Algebraic number fields and function fields over '''F'''<sub>''q''</sub> are further global fields. Studying arithmetic questions in global fields may sometimes be done by looking at the corresponding questions locally—this technique is called [[local-global principle]].
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| ==Some first theorems==<!-- This section is linked from [[Finite field]] -->
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| *Every finite subgroup of the multiplicative group ''F''<sup>×</sup> is [[cyclic group|cyclic]]. This applies in particular to '''F'''<sub>''q''</sub><sup>×</sup>, it is cyclic of order {{nowrap|''q'' − 1}}. In the introductory example, a [[generating set of a group|generator]] of '''F'''<sub>4</sub><sup>×</sup> is the element A.
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| *From the point of view of [[algebraic geometry]], fields are points, because the [[spectrum of a ring|spectrum]] ''Spec F'' has only one point, corresponding to the 0-ideal. This entails that a [[commutative ring]] is a field if and only if it has no [[Ideal (ring theory)|ideal]]s except {0} and itself. Equivalently, an integral domain is field if and only if its [[Krull dimension]] is 0.
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| *[[Isomorphism extension theorem]]
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| ==Constructing fields==
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| ===Closure operations===
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| Assuming the [[axiom of choice]], for every field ''F'', there exists a field {{Overline|''F''}}, called the [[algebraic closure]] of ''F'', which contains ''F'', is [[algebraic extension|algebraic]] over ''F'', which means that any element ''x'' of {{Overline|''F''}} satisfies a polynomial equation
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| :''f''<sub>''n''</sub>''x''<sup>''n''</sup> + ''f''<sub>''n''−1</sub>''x''<sup>''n''−1</sup> + ··· + ''f''<sub>1</sub>''x'' + ''f''<sub>0</sub> = 0, with coefficients ''f''<sub>''n''</sub>, ..., ''f''<sub>0</sub> ∈ ''F'',
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| and is [[algebraically closed]], i.e., any such polynomial does have at least one solution in {{Overline|''F''}}. The algebraic closure is unique [[up to]] isomorphism inducing the identity on ''F''. However, in many circumstances in mathematics, it is not appropriate to treat {{Overline|''F''}} as being uniquely determined by ''F'', since the isomorphism above is not itself unique. In these cases, one refers to such a {{Overline|''F''}} as ''an'' algebraic closure of ''F''. A similar concept is the [[separable closure]], containing all roots of [[separable polynomial]]s, instead of all polynomials.
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| For example, if {{nowrap begin}}''F'' = '''Q''',{{nowrap end}} the algebraic closure {{Overline|'''Q'''}} is also called ''field of [[algebraic number]]s''. The field of algebraic numbers is an example of an [[algebraically closed field]] of characteristic zero; as such it satisfies the same [[first-order sentence]]s as the field of complex numbers '''C'''.
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| In general, all algebraic closures of a field are isomorphic. However, there is in general no preferable isomorphism between two closures. Likewise for separable closures.
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| ===Subfields and field extensions===
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| A ''subfield'' is, informally, a small field contained in a bigger one. Formally, a subfield ''E'' of a field ''F'' is a [[subset]] containing 0 and 1, closed under the operations +, −, · and multiplicative inverses and with its own operations defined by restriction. For example, the real numbers contain several interesting subfields: the real [[algebraic number]]s, the [[computable number]]s and the [[rational number]]s are examples.
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| The notion of [[field extension]] lies at the heart of field theory, and is crucial to many other algebraic domains. A field extension ''F'' / ''E'' is simply a field ''F'' and a subfield ''E'' ⊂ ''F''. Constructing such a field extension ''F'' / ''E'' can be done by "adding new elements" or ''adjoining elements'' to the field ''E''. For example, given a field ''E'', the set ''F'' = ''E''(''X'') of [[rational function]]s, i.e., equivalence classes of expressions of the kind
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| :<math>\frac{p(X)}{q(X)},</math>
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| where ''p''(''X'') and ''q''(''X'') are polynomials with coefficients in ''E'', and ''q'' is not the zero polynomial, forms a field. This is the simplest example of a [[transcendental extension]] of ''E''. It also is an example of a [[domain (ring theory)|domain]] (the [[ring of polynomials]] <math>\scriptstyle E</math> in this case) being embedded into its [[field of fractions]] <math>\scriptstyle E(X)</math>.
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| The ring of [[formal power series]] <math>\scriptstyle E[[X]]</math> is also a domain, and again the (equivalence classes of) fractions of the form ''p''(''X'')/ ''q''(''X'') where ''p'' and ''q'' are elements of <math>\scriptstyle E[[X]]</math> form the field of fractions for <math>\scriptstyle E[[X]]</math>. This field is actually the ring of [[Laurent series]] over the field ''E'', denoted <math>\scriptstyle E((X))</math>.
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| In the above two cases, the added symbol ''X'' and its powers did not interact with elements of ''E''. It is possible however that the adjoined symbol may interact with ''E''. This idea will be illustrated by adjoining an element to the field of real numbers '''R'''. As explained above, '''C''' is an extension of '''R'''. '''C''' can be obtained from '''R''' by adjoining the [[imaginary number|imaginary]] symbol i which satisfies i<sup>2</sup> = −1. The result is that '''R'''[i]='''C'''. This is different from adjoining the symbol ''X'' to '''R''', because in that case, the powers of ''X'' are all distinct objects, but here, i<sup>2</sup>=−1 is actually an element of '''R'''.
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| Another way to view this last example is to note that i is a [[zero of a function|zero]] of the polynomial ''p''(''X'') = ''X''<sup>2</sup> + 1. The quotient ring <math>\scriptstyle R[X]/(X^2 \,+\, 1)</math> can be mapped onto '''C''' using the map <math>\scriptstyle \overline{a \,+\, bX} \;\rightarrow\; a \,+\, ib</math>. Since the [[ideal (ring theory)|ideal]] (''X''<sup>2</sup>+1) is generated by a polynomial irreducible over '''R''', the ideal is maximal, hence the [[quotient ring]] is a field. This nonzero ring map from the quotient to '''C''' is necessarily an isomorphism of rings.
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| The above construction generalises to any [[irreducible polynomial]] in the [[polynomial ring]] ''E''[''X''], i.e., a polynomial ''p''(''X'') that cannot be written as a product of non-constant polynomials. The quotient ring {{nowrap begin}}''F'' = ''E''[''X''] / (''p''(''X'')),{{nowrap end}} is again a field.
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| Alternatively, constructing such field extensions can also be done, if a bigger container is already given. Suppose given a field ''E'', and a field ''G'' containing ''E'' as a subfield, for example ''G'' could be the algebraic closure of ''E''. Let ''x'' be an element of ''G'' not in ''E''. Then there is a smallest subfield of ''G'' containing ''E'' and ''x'', denoted ''F'' = ''E''(''x'') and called ''field extension F / E generated by x in G''.<ref>Jacobson (2009), p. 213</ref> Such extensions are also called ''[[simple extension]]s''. Many extensions are of this type; see the [[primitive element theorem]]. 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.
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| One distinguishes between extensions having various qualities. For example, an extension ''K'' of a field ''k'' is called ''algebraic'', if every element of ''K'' is a root of some polynomial with coefficients in ''k''. Otherwise, the extension is called ''transcendental''. The aim of [[Galois theory]] is the study of ''[[algebraic extension]]s'' of a field.
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| ===Rings vs fields===
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| Adding multiplicative inverses to an integral domain ''R'' yields the [[field of fractions]] of ''R''. For example, the field of fractions of the integers '''Z''' is just '''Q'''.
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| Also, the field ''F''(''X'') is the quotient field of the [[polynomial ring|ring of polynomials]] ''F''[''X'']. "Getting back" the ring from the field is sometimes possible; see [[discrete valuation ring]].
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| Another method to obtain a field from a commutative ring ''R'' is taking the quotient {{nowrap|''R'' / ''m''}}, where ''m'' is any [[maximal ideal]] of ''R''. The above construction of {{nowrap begin}}''F'' = ''E''[''X''] / (''p''(''X'')),{{nowrap end}} is an example, because the irreducibility of the polynomial ''p''(''X'') is equivalent to the maximality of the ideal generated by this polynomial. Another example are the finite fields {{nowrap begin}}'''F'''<sub>''p''</sub> = '''Z''' / ''p'''''Z'''.{{nowrap end}}
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| ===Ultraproducts===
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| If ''I'' is an index set, ''U'' is an [[ultrafilter]] on ''I'', and ''F''<sub>''i''</sub> is a field for every ''i'' in ''I'', the [[ultraproduct]] of the ''F''<sub>''i''</sub> with respect to ''U'' is a field.
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| For example, a non-principal ultraproduct of finite fields is a pseudo finite field; i.e., a [[Pseudo algebraically closed field|PAC field]] having exactly one extension of any degree.
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| This construction is important to the study of the elementary theory of finite fields.
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| ==Galois theory==
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| {{main|Galois theory}}
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| Galois theory aims to study the [[algebraic extension]]s of a field by studying the [[Symmetry group#Symmetry groups in general|symmetry]] in the arithmetic operations of addition and multiplication. The [[fundamental theorem of Galois theory]] shows that there is a strong relation between the structure of the symmetry group and the set of algebraic extensions.
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| In the case where ''F'' / ''E'' is a [[finite extension|finite]] ([[Galois extension|Galois]]) extension, Galois theory studies the algebraic extensions of ''E'' that are subfields of ''F''. Such fields are called [[Field extension|intermediate extensions]]. Specifically, the [[Galois group]] of ''F'' over ''E'', denoted Gal(''F''/''E''), is the group of [[field automorphism]]s of ''F'' that are trivial on ''E'' (i.e., the [[bijection]]s σ : ''F'' → ''F'' that preserve addition and multiplication and that send elements of ''E'' to themselves), and the fundamental theorem of Galois theory states that there is a [[one-to-one correspondence]] between [[subgroup]]s of Gal(''F''/''E'') and the set of intermediate extensions of the extension ''F''/''E''. The theorem, in fact, gives an explicit correspondence and further properties.
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| To study all ([[Separable extension|separable]]) algebraic extensions of ''E'' at once, one must consider the [[absolute Galois group]] of ''E'', defined as the Galois group of the [[separable closure]], ''E''<sup>sep</sup>, of ''E'' over ''E'' (i.e., Gal(''E''<sup>sep</sup>/''E''). It is possible that the degree of this extension is infinite (as in the case of ''E'' = '''Q'''). It is thus necessary to have a notion of Galois group for an infinite algebraic extension. The Galois group in this case is obtained as a "limit" (specifically an [[inverse limit]]) of the Galois groups of the finite Galois extensions of ''E''. In this way, it acquires a [[Topological space|topology]].<ref group="note">As an inverse limit of [[Discrete group|finite discrete groups]], it is equipped with the [[limit topology|profinite topology]], making it a [[Profinite group|profinite topological group]]</ref> The fundamental theorem of Galois theory can be generalized to the case of infinite Galois extensions by taking into consideration the topology of the Galois group, and in the case of ''E''<sup>sep</sup>/''E'' it states that there this a one-to-one correspondence between ''[[Closed subgroup|closed]]'' subgroups of Gal(''E''<sup>sep</sup>/''E'') and the set of all separable algebraic extensions of ''E'' (technically, one only obtains those separable algebraic extensions of ''E'' that occur as subfields of the ''chosen'' separable closure ''E''<sup>sep</sup>, but since all separable closures of ''E'' are [[isomorphic]], choosing a different separable closure would give the same Galois group and thus an "equivalent" set of algebraic extensions).
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| ==Generalizations==
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| There are also [[proper class]]es with field structure, which are sometimes called '''Fields''', with a capital F:
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| *The [[surreal number]]s form a Field containing the reals, and would be a field except for the fact that they are a proper class, not a set.
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| *The [[nimber]]s form a Field. The set of nimbers with birthday smaller than 2<sup>2<sup>''n''</sup></sup>, the nimbers with birthday smaller than any infinite [[cardinal number|cardinal]] are all examples of fields.
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| In a different direction, [[differential field]]s are fields equipped with a [[derivation (abstract algebra)|derivation]]. For example, the field '''R'''(''X''), together with the standard derivative of polynomials forms a differential field. These fields are central to [[differential Galois theory]]. [[Exponential field]]s, meanwhile, are fields equipped with an exponential function that provides a homomorphism between the additive and multiplicative groups within the field. The usual [[exponential function]] makes the real and complex numbers exponential fields, denoted '''R'''<sub>exp</sub> and '''C'''<sub>exp</sub> respectively.
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| Generalizing in a more categorical direction yields the [[field with one element]] and related objects.
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| ===Exponentiation===
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| One does not in general study generalizations of fields with ''three'' binary operations. The familiar addition/subtraction, multiplication/division, exponentiation/root-extraction operations from the natural numbers to the reals, each built up in terms of iteration of the last, mean that generalizing exponentiation as a binary operation is tempting, but has generally not proven fruitful; instead, an exponential field assumes a unary exponential function from the additive group to the multiplicative group, not a partially defined binary function. Note that the exponential operation of <math>\scriptstyle a^b</math> is neither associative nor commutative, nor has a unique inverse (<math>\scriptstyle \pm 2</math> are both square roots of 4, for instance), unlike addition and multiplication, and further is not defined for many pairs—for example, <math>\scriptstyle (-1)^{1/2} \;=\; \sqrt{-1}</math> does not define a single number. These all show that even for rational numbers exponentiation is not nearly as well-behaved as addition and multiplication, which is why one does not in general axiomatize exponentiation.
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| ==Applications==
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| The concept of a field is of use, for example, in defining [[vector space|vector]]s and [[matrix (mathematics)|matrices]], two structures in [[linear algebra]] whose components can be elements of an arbitrary field.
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| [[Finite field]]s are used in [[number theory]], [[Galois theory]], [[coding theory]] and [[combinatorics]]; and again the notion of algebraic extension is an important tool.
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| Fields of [[characteristic (algebra)|characteristic]] 2 are useful in [[computer science]].
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| ==See also==
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| <div style="-moz-column-count:3; column-count:3;">
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| * [[Category of fields]]
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| * [[Glossary of field theory]] for more definitions in field theory.
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| * [[Heyting field]]
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| * [[Lefschetz principle]]
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| * [[Puiseux series]]
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| * [[ring (mathematics)|Ring]]
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| * [[Vector space]]
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| * [[Vector spaces without fields]]
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| </div>
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| ==Notes==
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| {{reflist|group=note}}
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| ==References==
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| {{More footnotes|date=March 2009}}
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| {{reflist}}
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| {{refbegin}}
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| * {{Citation | last1=Artin | first1=Michael | author1-link=Michael Artin | title=Algebra | publisher=[[Prentice Hall]] | isbn=978-0-13-004763-2 | year=1991}}, especially Chapter 13
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| * {{Citation | last1=Allenby | first1=R.B.J.T. | title=Rings, Fields and Groups | publisher=Butterworth-Heinemann | isbn=978-0-340-54440-2 | year=1991}}
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| * {{Citation | last1=Blyth | first1=T.S. | last2=Robertson | first2=E. F. | title=Groups, rings and fields: Algebra through practice| publisher=[[Cambridge University Press]] | year=1985}}. See especially Book 3 (ISBN 0-521-27288-2) and Book 6 (ISBN 0-521-27291-2).
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| * {{Citation| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| volume = 1 | series= | publisher=Dover| isbn = 978-0-486-47189-1}}
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| * James Ax (1968), ''The elementary theory of finite fields'', Ann. of Math. (2), '''88''', 239–271
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| {{refend}}
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| ==External links==
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| {{Wikibooks|Abstract algebra|Fields}}
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| * {{springer|title=Field|id=p/f040090}}
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| * [http://www.compsoc.nuigalway.ie/~pappasmurf/fields/index.php Field Theory Q&A]
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| *[http://www.apronus.com/provenmath/fields.htm Fields at ProvenMath] definition and basic properties.
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| *{{planetmath reference|id=355|title=Field}}
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| {{DEFAULTSORT:Field (Mathematics)}}
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| [[Category:Field theory]]
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| [[Category:Algebraic structures]]
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| [[ar:حقل رياضي]]
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| [[es:Cuerpo (matemática)]]
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| [[eo:Kampo (algebro)]]
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