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{{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 <math>n</math> variables over a [[field (mathematics)|field]] <math>k</math> is a finitely generated [[field extension]] <math>K/k</math> which has [[transcendence degree]] <math>n</math> over <math>k</math>.<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 <math>n</math> variables over <math>k</math> may be defined as a [[finite field extension]] of the field <math>K=k(x_1,\dots,x_n)</math> of [[rational functions]] in <math>n</math> variables over <math>k</math>.

==Example==
As an example, in the [[polynomial ring]] <math>k[x,y]</math> consider the [[ideal (ring theory)|ideal]] generated by the [[irreducible polynomial]] <math>y^2-x^3</math> and form the [[field of fractions]] of the [[quotient ring]] <math>k[x,y]/(y^2-x^3)</math>. This is a function field of one variable over <math>k</math>; 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 <math>k</math> form a [[category (mathematics)|category]]; the [[Morphism (category theory)|morphisms]] from function field <math>K</math> to <math>L</math> are the [[ring homomorphism]]s <math>f:K\to L</math> with <math>f(a)=a</math> for all <math>a</math> in <math>k</math>. All these morphisms are [[injective function|injective]]. If <math>K</math> is a function field over <math>k</math> of <math>n</math> variables, and <math>L</math> is a function field in <math>m</math> variables, and <math>n>m</math>, then there are no morphisms from <math>K</math> to <math>L</math>.

==Function fields arising from varieties, curves and Riemann surfaces==
The [[function field of an algebraic variety]] of dimension <math>n</math> over <math>k</math> is an algebraic function field of <math>n</math> variables over <math>k</math>.
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 <math>k</math> (with [[rational mapping|dominant rational maps]] as morphisms) and the category of algebraic function fields over <math>k</math>. The varieties considered here are to be taken in the [[scheme (mathematics)|scheme]] sense; they need not have any <math>k</math>-rational points, like the curve <math>x^2+y^2+1=0</math> defined over the real numbers.

The case <math>n=1</math> (irreducible algebraic curves in the [[scheme (mathematics)|scheme]] sense) is especially important, since every function field of one variable over <math>k</math> arises as the function field of a uniquely defined [[regular scheme|regular]] (i.e. non-singular) projective irreducible algebraic curve over <math>k</math>. 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 <math>k</math>.

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

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

For instance, <math>\C(x)</math> is a function field of one variable over <math>\R</math>; its field of constants is <math>\C</math>.

==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 <math>K/k</math> of one variable, we define the notion of a ''valuation ring'' of <math>K/k</math>: this is a [[subring]] <math>\mathcal{O}</math> of <math>K</math> that contains <math>k</math> and is different from <math>k</math> and <math>K</math>, and such that for any <math>x</math> in <math>K</math> we have <math>x\in\mathcal{O}</math> or <math>x^{-1}\in\mathcal{O}</math>. Each such valuation ring is a [[discrete valuation ring]] and its maximal ideal is called a ''place'' of <math>K/k</math>.

A ''discrete valuation'' of <math>K/k</math> is a [[surjective]] function <math>v:K\to\Z\cup\{\infty\}</math> such that for all <math>x,y\in K</math>,
* <math>v(xy)=v(x)+v(y)</math>,
* <math>v(x+y)\geq \min\{v(x),v(y)\}</math>,
* <math>v(x)=\infty \iff x=0</math>
and <math>v(a)=0</math> for all <math>a\in k\setminus \{0\}</math>.

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

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

==References==
{{reflist}}

[[Category:Field (mathematics)]]

Algebraic function field - Revision history

imported>Barçaforlife at 23:58, 25 June 2025