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}.