Let $A \subseteq B$ be a finite extension of Dedekind domains such that the extension $K \subseteq L$ of their quotient fields is separable. Let $\mathfrak{p}$ be a maximal ideal of $A$ and let $\mathfrak{q}_1, \dots, \mathfrak{q}_r$ be the primes of $B$ that lie over $\mathfrak{p}$. Let $e_i$ be the ramification index of $\mathfrak{q}_i$ over $\mathfrak{p}$.
Suppose $e_1 \geq 2$ and consider the ideal $I = \mathfrak{q}_1 \mathfrak{q}_2^{e_2} \cdots \mathfrak{q}_r^{e_r}$. I'd like to have a proof of the following containment: $ \mathfrak{p} \supseteq \mathrm{Tr}_{L/K} (I), $ where $\mathrm{Tr}_{L/K} \colon L \to K$ is the trace of the field extension $L \supseteq K$. This fact is Exercise 37-(a) at page 95 of Marcus' Number fields.
Thanks to all!