Integral basis of an extension of number fields

Let $K\subseteq F$ be number fields with ring of integers $\mathcal{O}_K\le \mathcal{O}_F$.

Question: Is $\mathcal{O}_F$ a free $\mathcal{O}_K$-module ?

By the integral basis theorem this is true when $K=\mathbb{Q}$ but I don't know about the general case.

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No. The following notes from Keith Conrad's website give a family of examples with $\mathscr{O}_F$ not $\mathscr{O}_K$-free: