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 $L$ .