Editing Field (mathematics) (section)

=== 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.