Is $O_L$ a free $O_K$ module?

Question

This must have been known. Let $L/K$ be a finite number field extension. Then is $O_L$ a free $O_K$ module? How to prove it if so? So far, I know how to prove the following: Let $\alpha_1,\dotsc,\alpha_n\in O_L$ be a $K$-basis of $L$ and let $d=d_{L/K}(\alpha_1,\dotsc,\alpha_n)$. Then $dO_L\subset O_K\alpha_1\oplus\dotsm\oplus O_K\alpha_n$.

Answer 1

No. This question has been answered before here by linking to these notes from Keith Conrad.