Find an extension $L/K$ of number fields with Galois group $G$ and respective rings of integers $O_L$ and $O_K$ for each of the following requirements:
The decomposition group $G_q$ of some prime ideal $q$ of $O_L$ over $p = q \cap O_K$ is not a normal subgroup of $G$.
$G=I_q\times I_{q'}$ is the direct product of two nontrivial inertia subgroups $I_q$ and $I_{q'}$, where $q, q'$ are prime ideals of $O_L$
The inertia group of $I_q$ is not cyclic for a prime ideal $q$ of $O_L$.
The only examples I know how to work with (like $\mathbb{Q}(i)$, $\mathbb{Q}(\sqrt[3]{2})$ or simple cyclotomic extensions) apperently are not enough for this exercise. Is there some strategy to find these examples?