Rings of integers of number field and free modules over a PID
If $K \subseteq L$ are number fields and $O_{L}, O_{K}$ are their ring of integers, how can I prove that there exists a basis of $O_{L}$ over $\mathbb{Z}$ that contains a basis of $O_{K}$ over $\mathbb{Z}$? I think it is trivial if $O_{L}$ is free over $O_K$, but is it true in general?
Solution 1:
$O_L/O_K$ is a torsion-free finitely generated abelian group (because $O_K=O_L \cap K$), and in particular the exact sequence of abelian groups $0 \rightarrow O_K \rightarrow O_L \rightarrow O_L/O_K \rightarrow 0$ splits, and $O_L \cong O_K \oplus O_L/O_K$. Hence the conclusion.