# Riemann-Hurwitz theorem

First we define the different divisor of an extension of function fields. Let $K$ be a function field over a field $F$ and let $L$ be a finite separable extension of $K$. Let $\mathcal{O}_{P}$ be a prime of $K$, i.e. a discrete valuation ring with $F\subset\mathcal{O}_{P}$, maximal ideal $P$ and quotient field equal to $K$. Let $R_{P}$ be the integral closure of $\mathcal{O}_{P}$ in $L$. Notice that if $\mathfrak{p}$ is a prime ideal of $R_{P}$, then the localization $\mathcal{O}_{\mathfrak{p}}=(R_{P})_{\mathfrak{p}}$ is a prime of $L$ (which is said to be lying over $\mathcal{O}_{P}$). The maximal ideal of $\mathcal{O}_{\mathfrak{p}}$ is $\mathfrak{p}(R_{P})_{\mathfrak{p}}$.

Let $\mathcal{O}_{\mathfrak{P}}$ be any prime of $L$, then it lays over some prime ideal $P$ of $K$ and in fact, if $\mathfrak{p}=R_{P}\cap\mathfrak{P}$ then $\mathcal{O}_{\mathfrak{p}}\cong\mathcal{O}_{\mathfrak{P}}$. Let $\delta(\mathfrak{P})$ be the exact power of $\mathfrak{p}$ dividing the different of $R_{P}$ over $\mathcal{O}_{P}$ (the different of an extension of Dedekind domains is a fractional ideal). We define the different divisor of $L/K$ as follows:

 $D_{L/K}=\sum_{\mathfrak{P}}\delta({\mathfrak{P}})\mathfrak{P}$

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 $g_{K}$. Then the genus of $L$ is given by the formula:

 $2g_{L}-2=[L:K](2g_{K}-2)+\deg_{L}D_{L/K}.$
Title Riemann-Hurwitz theorem RiemannHurwitzTheorem 2013-03-22 15:34:40 2013-03-22 15:34:40 alozano (2414) alozano (2414) 4 alozano (2414) Theorem msc 11R58 different divisor of an extension of function fields