function field

Let F be a field.

Definition 1.

The rational function fieldPlanetmathPlanetmath over F in one variable (x), denoted by F(x), is the field of all rational functions p(x)/q(x) with polynomialsMathworldPlanetmathPlanetmathPlanetmath p,qF[x] and q(x) not identically zero.

Definition 2.

A function field (in one variable) over F is a field K, containing F and at least one element x, transcendentalPlanetmathPlanetmath over F, such that K/F(x) is a algebraic extensionMathworldPlanetmath.

Let F¯ be a fixed algebraic closureMathworldPlanetmath of F.

Definition 3.

Let K be a function field over F and let L be a finite extensionMathworldPlanetmath of K. The extension L/K of function fields is said to be geometric if LF¯=F.

Example 1.

The extension (x)/(x) is geometric, but (2)(x)/(x) is not geometric.

Theorem 1 (Thm. I.6.9 of [1]).

Let K be a function field over an algebraically closed field F. There exists a nonsingular projective curve CK such that the function field of CK is isomorphic to K.

Definition 4.

Let K be a function field over a field F. Let K=KF¯ which is a function field over F¯, a fixed algebraic closure of F, and let CK be the curve given by the previous theorem. The genus of K is, by definition, the genus of CK.

Definition 5.

Let K be a function field over a field F. A prime in K is by definition a discrete valuation ring R with maximal P such that FR and the quotient field of R is equal to K. The prime is usually denoted P after the maximal idealMathworldPlanetmath of R. The degree of P, denoted by degP, is defined to be the dimensionMathworldPlanetmath of R/P over F.

Example 2.

Let K=F(x) be the rational function field over F and let 𝒪=F[x]. The prime idealsMathworldPlanetmathPlanetmath of 𝒪 are generated by monic irreducible polynomialsMathworldPlanetmath in F[x]. Let P=(f(x)) be such a prime. Then RP=𝒪P, the localizationMathworldPlanetmath of 𝒪 at the prime P is a discrete valuation ring with F𝒪P and the quotient field of RP is K. Thus RP=𝒪P is a prime of K.

One can define an ‘extra’ prime in the following way. Let R=𝒪=F[1x] and let P=(1x) be the prime ideal of R generated by 1x. The localization ring (R)P is a prime of K, called the prime at infinity.


Title function field
Canonical name FunctionField
Date of creation 2013-03-22 15:34:35
Last modified on 2013-03-22 15:34:35
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 8
Author alozano (2414)
Entry type Definition
Classification msc 11R58
Synonym algebraic function field
Related topic SimpleTranscendentalFieldExtension
Defines rational function field
Defines geometric extension
Defines genus of a function field
Defines degree of a prime