Let $K$ be an algebraic function field over $\mathbb F_q$, in other words $K$ is a finite extension of $\mathbb F_q(t)=F$; furthermore fix a valuation $v$ on $K$.
Consider the relative different ideal $\mathfrak D_v$ with respect to $K|F$. It tourns out that $\mathfrak D$ is a principal ideal of the valuation ring $\mathcal O_v$, so suppose that $\mathfrak D=d\mathcal O_v$. I would like to know a way to calculate $|d|_v$ where $|\cdot|_v$ is the absolute value associated to $v$.
Many sources (mainly texts on Tate's thesis) claim that $|d_v|=\frac{1}{\#(\mathcal O_v/\mathfrak D_v)}$, but I don't understand why this is true. Also it is not clear why $\#(\mathcal O_v/\mathfrak D_v)$ is finite.