Let $D$ be a Dedekind domain. Let $v:D \to \mathbb{R}$ a valuation. We know that for every prime ideal $\mathfrak p$ of $D$ the localization $D_{\mathfrak p}$ is DVR.
Does every valuation on $D$ directly arise from some valuation on some $D_{\mathfrak p}$ ?
How can we find all the valuations $v:D \to \mathbb{R}$ ?
If $D$ is the ring of integers of some number field, then the answer is yes by Ostrowski's theorem over $\mathbb Q$.
Otherwise, even for function fields, not all (non-Archimedean) valuations are discrete, so not all of them come from a localization $D_p$. For example, fix an irrational $\theta\in \mathbb R$ and a prime number $p$, and consider the Gauss valuation $$ \sum_{n\ge 0} a_n t^n \mapsto \min_n \{ v_p(a_n)+n\theta \}$$ on $\mathbb Q[t]$. Its values group $\mathbb Z+\theta \mathbb Z$ is dense in $\mathbb R$.