Riemann-Hurwitz theorem

First we define the different divisorMathworldPlanetmathPlanetmath of an extension of function fields. Let K be a function fieldMathworldPlanetmath over a field F and let L be a finite separable extensionMathworldPlanetmath of K. Let 𝒪P be a prime of K, i.e. a discrete valuation ring with F𝒪P, maximal idealMathworldPlanetmath P and quotient field equal to K. Let RP be the integral closureMathworldPlanetmath of 𝒪P in L. Notice that if 𝔭 is a prime idealMathworldPlanetmathPlanetmath of RP, then the localizationMathworldPlanetmath 𝒪𝔭=(RP)𝔭 is a prime of L (which is said to be lying over 𝒪P). The maximal ideal of 𝒪𝔭 is 𝔭(RP)𝔭.

Let 𝒪𝔓 be any prime of L, then it lays over some prime ideal P of K and in fact, if 𝔭=RP𝔓 then 𝒪𝔭𝒪𝔓. Let δ(𝔓) be the exact power of 𝔭 dividing the different of RP over 𝒪P (the different of an extensionPlanetmathPlanetmath of Dedekind domainsMathworldPlanetmath is a fractional idealMathworldPlanetmathPlanetmath). We define the different divisor of L/K as follows:


as an element of the free abelian group generated by the prime ideals of L.

Theorem (Riemann-Hurwitz).

Let L/K be a finite, separable, geometric extension of function fields and suppose the genus of K is gK. Then the genus of L is given by the formulaMathworldPlanetmathPlanetmath:

Title Riemann-Hurwitz theorem
Canonical name RiemannHurwitzTheorem
Date of creation 2013-03-22 15:34:40
Last modified on 2013-03-22 15:34:40
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 4
Author alozano (2414)
Entry type Theorem
Classification msc 11R58
Defines different divisor of an extension of function fields