Editing Field (mathematics) (section)

== Applications ==

=== Linear algebra and commutative algebra ===
If {{math|''a'' ≠ 0}}, then the [[equation]]
: {{math|1=''ax'' = ''b''}}
has a unique solution {{math|''x''}} in a field {{math|''F''}}, namely <math>x=a^{-1}b.</math> This immediate consequence of the definition of a field is fundamental in [[linear algebra]]. For example, it is an essential ingredient of [[Gaussian elimination]] and of the proof that any [[vector space]] has a [[basis (linear algebra)|basis]].<ref>{{harvp|Artin|1991|loc=§3.3}}</ref>

The theory of [[module (mathematics)|modules]] (the analogue of vector spaces over [[ring (mathematics)|ring]]s instead of fields) is much more complicated, because the above equation may have several or no solutions. In particular [[linear equation over a ring|systems of linear equations over a ring]] are much more difficult to solve than in the case of fields, even in the specially simple case of the ring {{math|'''Z'''}} of the integers.

=== Finite fields: cryptography and coding theory ===
[[File:ECClines.svg|thumb|The sum of three points {{math|''P''}}, {{math|''Q''}}, and {{math|''R''}} on an elliptic curve {{math|''E''}} (red) is zero if there is a line (blue) passing through these points.]]
A widely applied cryptographic routine uses the fact that discrete exponentiation, i.e., computing
: {{math|1=''a''<sup>''n''</sup> = ''a'' ⋅ ''a'' ⋅ ⋯ ⋅ ''a''}} ({{math|''n''}} factors, for an integer {{math|''n'' ≥ 1}})
in a (large) finite field {{math|'''F'''<sub>''q''</sub>}} can be performed much more efficiently than the [[discrete logarithm]], which is the inverse operation, i.e., determining the solution {{math|''n''}} to an equation
: {{math|1=''a''<sup>''n''</sup> = ''b''}}.
In [[elliptic curve cryptography]], the multiplication in a finite field is replaced by the operation of adding points on an [[elliptic curve]], i.e., the solutions of an equation of the form
: {{math|1=''y''<sup>2</sup> = ''x''<sup>3</sup> + ''ax'' + ''b''}}.

Finite fields are also used in [[coding theory]] and [[combinatorics]].

=== Geometry: field of functions ===
[[File:Double torus illustration.png|thumb|A compact Riemann surface of [[genus (mathematics)|genus]] two (two handles). The genus can be read off the field of meromorphic functions on the surface.]]
[[function (mathematics)|Functions]] on a suitable [[topological space]] {{math|''X''}} into a field {{mvar|F}} can be added and multiplied pointwise, e.g., the product of two functions is defined by the product of their values within the domain:
: {{math|1={{nowrap|1=(''f'' ⋅ ''g'')(''x'') = ''f''(''x'') ⋅ ''g''(''x'')}}}}.
This makes these functions a {{math|''F''}}-[[associative algebra|commutative algebra]].

For having a ''field'' of functions, one must consider algebras of functions that are [[integral domains]]. In this case the ratios of two functions, i.e., expressions of the form
: <math>\frac{f(x)}{g(x)},</math>
form a field, called field of functions.

This occurs in two main cases. When {{math|''X''}} is a [[complex manifold]] {{math|''X''}}. In this case, one considers the algebra of [[holomorphic functions]], i.e., complex differentiable functions. Their ratios form the field of [[meromorphic function]]s on {{math|''X''}}.

The [[function field of an algebraic variety]] {{math|''X''}} (a geometric object defined as the common zeros of polynomial equations) consists of ratios of [[regular function]]s, i.e., ratios of polynomial functions on the variety. The function field of the {{math|''n''}}-dimensional [[affine space|space]] over a field {{math|''F''}} is {{math|''F''(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>)}}, i.e., the field consisting of ratios of polynomials in {{math|''n''}} indeterminates. The function field of {{math|''X''}} is the same as the one of any [[Zariski topology|open]] dense subvariety. In other words, the function field is insensitive to replacing {{math|''X''}} by a (slightly) smaller subvariety.

The function field is invariant under [[isomorphism]] and [[birational equivalence]] of varieties. It is therefore an important tool for the study of [[abstract algebraic variety|abstract algebraic varieties]] and for the classification of algebraic varieties. For example, the [[dimension of an algebraic variety|dimension]], which equals the transcendence degree of {{math|''F''(''X'')}}, is invariant under birational equivalence.<ref>{{harvp|Eisenbud|1995|loc=§13, Theorem A}}</ref> For [[algebraic curve|curves]] (i.e., the dimension is one), the function field {{math|''F''(''X'')}} is very close to {{math|''X''}}: if {{math|''X''}} is [[smooth variety|smooth]] and [[proper map|proper]] (the analogue of being [[compact topological space|compact]]), {{math|''X''}} can be reconstructed, up to isomorphism, from its field of functions.{{efn|More precisely, there is an [[equivalence of categories]] between smooth proper algebraic curves over an algebraically closed field {{math|''F''}} and finite field extensions of {{math|''F''(''T'')}}.}} In higher dimension the function field remembers less, but still decisive information about {{math|''X''}}. The study of function fields and their geometric meaning in higher dimensions is referred to as [[birational geometry]]. The [[minimal model program]] attempts to identify the simplest (in a certain precise sense) algebraic varieties with a prescribed function field.

=== Number theory: global fields ===
[[Global field]]s are in the limelight in [[algebraic number theory]] and [[arithmetic geometry]].
They are, by definition, [[number field]]s (finite extensions of {{math|'''Q'''}}) or function fields over {{math|'''F'''<sub>''q''</sub>}} (finite extensions of {{math|'''F'''<sub>''q''</sub>(''t'')}}). As for local fields, these two types of fields share several similar features, even though they are of characteristic {{math|0}} and positive characteristic, respectively. This [[function field analogy]] can help to shape mathematical expectations, often first by understanding questions about function fields, and later treating the number field case. The latter is often more difficult. For example, the [[Riemann hypothesis]] concerning the zeros of the [[Riemann zeta function]] (open as of 2017) can be regarded as being parallel to the [[Weil conjectures]] (proven in 1974 by [[Pierre Deligne]]).

[[File:One5Root.svg|thumb|The fifth roots of unity form a [[regular pentagon]].]]
[[Cyclotomic field]]s are among the most intensely studied number fields. They are of the form {{math|'''Q'''(''ζ''<sub>''n''</sub>)}}, where {{math|''ζ''<sub>''n''</sub>}} is a primitive {{math|''n''}}th [[root of unity]], i.e., a complex number {{math|''ζ''}} that satisfies {{math|1={{itco|''ζ''}}<sup>''n''</sup> = 1}} and {{math|{{itco|''ζ''}}<sup>''m''</sup> ≠ 1}} for all {{math|0 < ''m'' < ''n''}}.<ref>{{harvp|Washington|1997}}</ref> For {{math|''n''}} being a [[regular prime]], [[Ernst Kummer|Kummer]] used cyclotomic fields to prove [[Fermat's Last Theorem]], which asserts the non-existence of rational nonzero solutions to the equation
: {{math|1=''x''<sup>''n''</sup> + ''y''<sup>''n''</sup> = ''z''<sup>''n''</sup>}}.

Local fields are completions of global fields. [[Ostrowski's theorem]] asserts that the only completions of {{math|'''Q'''}}, a global field, are the local fields {{math|'''Q'''<sub>''p''</sub>}} and {{math|'''R'''}}. Studying arithmetic questions in global fields may sometimes be done by looking at the corresponding questions locally. This technique is called the [[local–global principle]]. For example, the [[Hasse–Minkowski theorem]] reduces the problem of finding rational solutions of quadratic equations to solving these equations in {{math|'''R'''}} and {{math|'''Q'''<sub>''p''</sub>}}, whose solutions can easily be described.<ref>{{harvp|Serre|1996|loc=Chapter IV}}</ref>

Unlike for local fields, the Galois groups of global fields are not known. [[Inverse Galois theory]] studies the (unsolved) problem whether any finite group is the Galois group {{math|Gal(''F''/'''Q''')}} for some number field {{math|''F''}}.<ref>{{harvp|Serre|1992}}</ref> [[Class field theory]] describes the [[abelian extension]]s, i.e., ones with abelian Galois group, or equivalently the abelianized Galois groups of global fields. A classical statement, the [[Kronecker–Weber theorem]], describes the maximal abelian {{math|'''Q'''<sup>ab</sup>}} extension of {{math|'''Q'''}}: it is the field
: {{math|'''Q'''(''ζ''<sub>''n''</sub>, ''n'' ≥ 2)}}
obtained by adjoining all primitive {{math|''n''}}th roots of unity. [[Kronecker Jugendtraum|Kronecker's Jugendtraum]] asks for a similarly explicit description of {{math|''F''<sup>ab</sup>}} of general number fields {{math|''F''}}. For [[imaginary quadratic field]]s, <math>F=\mathbf Q(\sqrt{-d})</math>, {{math|''d'' > 0}}, the theory of [[complex multiplication]] describes {{math|''F''<sup>ab</sup>}} using [[elliptic curves]]. For general number fields, no such explicit description is known.