Exercise 10, § 1

Let $A$ be a Krull domain, $K$ its field of fractions, $L$ a field containing $K$ and $(B_i)$ a right directed family of Krull domains containing $A$ and contained in $L$ such that the field of fractions $L_\alpha$ of $B_i$ is a finite algebraic extension of $K$ and $B_i$ is the integral closure of $A$ in $L_\alpha$. For every essential valuation $v$ of $A$ and all $a$, let $e_\alpha(v)$ be the sum of the ramification indices of the valuations on $B_i$ which extend $v$. Show that for the union of the $B_i$ to be a Krull domain, it is necessary and sufficient that for every essential valuation $v$ of $A$, the set of $e_\alpha(v)$ be bounded.