Let $L/K$ be a field extension of number fields such that the class number of $K$ is greater one (i.e. $\mathcal O_K$ is not a principal ideal domain).
Are there examples where $O_L$ is not free over $O_K$? If yes: What is the smallest such example?
Or even better: Is there for every $K$ such an $L$?
One standard example is the number field $K=\Bbb{Q}(\sqrt{-14})$ with ring of integers $\mathcal{O}_K=\Bbb{Z}[\sqrt{-14}]$, which is not a PID. Then the extension of number fields $L/K$ with $L=\Bbb{Q}(\sqrt{-14},\sqrt{-7})$ has the property that $\mathcal{O}_L$ is not free over $\mathcal{O}_K$.
Reference: The book "Number Theory" by Narkewicz. He shows that whenever $\mathcal{O}_K$ is not a PID there exists a quadratic extension $L/K$ such that $\mathcal{O}_L$ is not free as an $\mathcal{O}_K$-module.