Definition 9.6 (Valuation on a Dedekind domain). If $R$ is a Dedekind domain, and $p\subseteq R$ a non-zero prime ideal, we write $v_p$ for the normalised valuation on $\mathrm{Frac}(R)=\mathrm{Frac}(R_{(p)})$ corresponding to the discrete valuation ring $R_{(p)}$.