Let $A$ be a Dedekind domain. Let $P$ be a non-zero prime ideal of $A$. Let $\alpha \in A$. Let $k$ be a non-negative integer. If $\alpha \in P^k$ and $\alpha\notin P^{k+1}$, we write $v_P(\alpha) = k$.
How can we prove the following
Proposition. Let $A$ be a Dedekind domain. Let $P_1,\dots, P_n$ be distinct non-zero prime ideals of $A$. Let $e_1, \dots, e_n$ be non-negative integers. Then there exists $\alpha \in A$ such that $v_{P_i}(\alpha) = e_i$, $i = 1,\dots, n$.