Exercise $17$ on Marcus' Number Fields, Chapter $4$, has $9$ items (from $a$ to $i$). I got stuck at item $e$.
(The notation is the following: Whenever $L$ is a normal extension of $K$ and $H$ is a subgroup of $\text{Gal}\left (L/K\right )$, for every subset $X\subset L$ the set $X_H$ is the set of elements of $X$ fixed pointwise by $H$).
The second and the third part of the exercise are easy, because we are supposing that $Q$ is unramified, therefore if $PS=QI$ then $Q$ is coprime to $I$ and hence $Q+I=S$. If the first part were true, the third part follows immediately, so we only have to prove the first part.
I tried to use that $\sigma \left (t\right )\equiv t\pmod U$ for every $\sigma \in \text{Gal}\left (M/L\right )=E$ and then use the trace $T_{L}^M$, but this only leads to $\left | E\right | T\subseteq S+U$.
I also tried to use the fact that $U$ is a maximal ideal of $T$ and $S$ is not included in $U$, but then I could not prove that $S+U$ is an ideal of $T$.
Any help?