Exercise 11, § 2

Let $K$ be a field and $(v_i)_{i \in I}$ a family of discrete valuations on $K$ satisfying conditions (AK_I) and (AK_{III}) of § 1, no. 3 and such that, for every integer $r$, every family $(n_h)_{1 \leq h \leq r}$ of integers $\geq 0$, every family $(t_h)_{1 \leq h \leq r}$ of distinct elements of $I$ and every family $(a_h)_{1 \leq h \leq r}$ of elements of $K$, there exists $x \in K$ such that $v_{t_h}(x - a_h) \geq n_h$ for $1 \leq h \leq r$ and $v_i(x) = 0$ for $i$ distinct from the $t_h$. Show that the intersection $A$ of the valuation rings of the $v_i$, is a Dedekind domain whose field of fractions is $K$ and that $(v_i)_{i \in I}$ is the family of essential valuations of $A$. (Use Exercise 7 of § 1; then prove that two distinct prime ideals of height 1 are relatively prime and deduce that every prime ideal $\neq 0$ of $A$ is maximal.)