Editing Algebraic function field

{{short description|Finitely generated extension field of positive transcendence degree}}
{{refimprove|date=December 2021}}
In [[mathematics]], an '''algebraic function field''' (often abbreviated as '''function field''') of ''n'' variables over a [[field (mathematics)|field]] ''k'' is a finitely generated [[field extension]] ''K''/''k'' which has [[transcendence degree]] ''n'' over ''k''.<ref>{{cite book |author=Gabriel Daniel |author2=Villa Salvador |name-list-style=amp|title=Topics in the Theory of Algebraic Function Fields|publisher=Springer |year= 2007|isbn=9780817645151|url=https://books.google.com/books?id=RmKpEUltmQIC}}</ref> Equivalently, an algebraic function field of ''n'' variables over ''k'' may be defined as a [[finite field extension]] of the field ''K'' = ''k''(''x''<sub>1</sub>,...,''x''<sub>''n''</sub>) of [[rational functions]] in ''n'' variables over ''k''.

==Example==
As an example, in the [[polynomial ring]] ''k''{{space|hair}}[''X'',''Y''] consider the [[ideal (ring theory)|ideal]] generated by the [[irreducible polynomial]] ''Y''<sup>{{space|hair}}2</sup>&thinsp;−&thinsp;''X''<sup>{{space|hair}}3</sup> and form the [[field of fractions]] of the [[quotient ring]] ''k''{{space|hair}}[''X'',''Y'']/(''Y''<sup>{{space|hair}}2</sup>&thinsp;−&thinsp;''X''<sup>{{space|hair}}3</sup>). This is a function field of one variable over ''k''; it can also be written as <math>k(X)(\sqrt{X^3})</math> (with degree 2 over <math>k(X)</math>) or as <math>k(Y)(\sqrt[3]{Y^2})</math> (with degree 3 over <math>k(Y)</math>). We see that the degree of an algebraic function field is not a well-defined notion.

==Category structure==
The algebraic function fields over ''k'' form a [[category (mathematics)|category]]; the [[Morphism (category theory)|morphisms]] from function field ''K'' to ''L'' are the [[ring homomorphism]]s ''f''&nbsp;:&nbsp;''K'' → ''L'' with ''f''(''a'') = ''a'' for all ''a'' in ''k''. All these morphisms are [[injective function|injective]]. If ''K'' is a function field over ''k'' of ''n'' variables, and ''L'' is a function field in ''m'' variables, and ''n'' > ''m'', then there are no morphisms from ''K'' to ''L''.

==Function fields arising from varieties, curves and Riemann surfaces==
The [[function field of an algebraic variety]] of dimension ''n'' over ''k'' is an algebraic function field of ''n'' variables over ''k''.
Two varieties are [[birational geometry|birationally equivalent]] if and only if their function fields are isomorphic. (But note that non-[[morphism of varieties|isomorphic]] varieties may have the same function field!) Assigning to each variety its function field yields a [[equivalence of categories|duality]] (contravariant equivalence) between the category of varieties over ''k'' (with [[rational mapping|dominant rational maps]] as morphisms) and the category of algebraic function fields over ''k''. (The varieties considered here are to be taken in the [[scheme (mathematics)|scheme]] sense; they need not have any ''k''-rational points, like the curve {{math|1=''X''<sup>2</sup> + ''Y''<sup>2</sup> + 1 = 0}} defined over the [[Real number|reals]], that is with {{math|1=''k'' = '''R'''}}.)

The case ''n''&thinsp;=&thinsp;1 (irreducible algebraic curves in the [[scheme (mathematics)|scheme]] sense) is especially important, since every function field of one variable over ''k'' arises as the function field of a uniquely defined [[regular scheme|regular]] (i.e. non-singular) projective irreducible algebraic curve over ''k''. In fact, the function field yields a duality between the category of regular projective irreducible algebraic curves (with [[Glossary of scheme theory#dominant|dominant]] [[regular map (algebraic geometry)|regular map]]s as morphisms) and the category of function fields of one variable over ''k''.

The field M(''X'') of [[meromorphic function]]s defined on a connected [[Riemann surface]] ''X'' is a function field of one variable over the [[complex number]]s '''C'''. In fact, M yields a duality (contravariant equivalence) between the category of compact connected Riemann surfaces (with non-constant [[holomorphic]] maps as morphisms) and function fields of one variable over '''C'''. A similar correspondence exists between compact connected [[Klein surface]]s and function fields in one variable over '''R'''.

==Number fields and finite fields==
The [[function field analogy]] states that almost all theorems on [[number field]]s have a counterpart on function fields of one variable over a [[finite field]], and these counterparts are frequently easier to prove. (For example, see [[Prime number theorem#Analogue for irreducible polynomials over a finite field|Analogue for irreducible polynomials over a finite field]].) In the context of this analogy, both number fields and function fields over finite fields are usually called "[[global field]]s".

The study of function fields over a finite field has applications in [[cryptography]] and [[error correcting code]]s. For example, the function field of an [[elliptic curve]] over a finite field (an important mathematical tool for [[public key cryptography]]) is an algebraic function field.

Function fields over the field of [[rational number]]s play also an important role in solving [[inverse Galois problem]]s.

==Field of constants==
Given any algebraic function field ''K'' over ''k'', we can consider the [[Set (mathematics)|set]] of elements of ''K'' which are [[algebraic element|algebraic]] over ''k''. These elements form a field, known as the ''field of constants'' of the algebraic function field.

For instance, '''C'''(''x'') is a function field of one variable over '''R'''; its field of constants is '''C'''.

==Valuations and places==
Key tools to study algebraic function fields are [[absolute value (algebra)|absolute values, valuations, places]] and their completions.

Given an algebraic function field ''K''/''k'' of one variable, we define the notion of a ''valuation ring'' of ''K''/''k'': this is a [[subring]] ''O'' of ''K'' that contains ''k'' and is different from ''k'' and ''K'', and such that for any ''x'' in ''K'' we have ''x''&thinsp;&isin;&thinsp;''O'' or ''x''<sup>&nbsp;-1</sup>&thinsp;&isin;&thinsp;''O''. Each such valuation ring is a [[discrete valuation ring]] and its maximal ideal is called a ''place'' of ''K''/''k''.

A ''discrete valuation'' of ''K''/''k'' is a [[surjective]] function ''v'' : ''K'' → '''Z'''∪{&infin;} such that ''v''(x)&thinsp;=&thinsp;&infin; iff ''x''&thinsp;=&thinsp;0, ''v''(''xy'') = ''v''(''x'')&thinsp;+&thinsp;''v''(''y'') and ''v''(''x''&thinsp;+&thinsp;''y'') ≥ min(''v''(''x''),''v''(''y'')) for all ''x'',&thinsp;''y''&thinsp;&isin;&thinsp;''K'', and ''v''(''a'')&thinsp;=&thinsp;0 for all ''a''&thinsp;&isin;&thinsp;''k''&thinsp;\&thinsp;{0}.

There are natural bijective correspondences between the set of valuation rings of ''K''/''k'', the set of places of ''K''/''k'', and the set of discrete valuations of ''K''/''k''. These sets can be given a natural [[Topology|topological]] structure: the [[Zariski–Riemann space]] of ''K''/''k''.

==See also==
*[[function field of an algebraic variety]]
*[[function field (scheme theory)]]
*[[algebraic function]]
*[[Drinfeld module]]

==References==
{{reflist}}

[[Category:Field (mathematics)]]