Suppose that we have a number field $K$ and ring of integers $\mathcal{O}_K$. Let $\mathfrak{a} \subset \mathfrak{a}'$ be two finitely generated $\mathcal{O}_K$ submodules of $K$. Then the proposition states that $d(\mathfrak{a}) = (\mathfrak{a}' : \mathfrak{a}) d(\mathfrak{a}')$ where $d(\mathfrak{a})$ is the discriminant and $(\mathfrak{a}' : \mathfrak{a})$ is the index. Some online notes use this to show a useful corollary: If $K = \mathbb{Q}(\alpha)$ for some $\alpha \in \mathcal{O}_K$, then we have $$ d(1, \alpha, \ldots, \alpha^{n - 1}) = (\mathcal{O}_K : \mathbb{Z}[\alpha])^2 \Delta_K.$$
However, I don't see how this can be applied, as $\mathbb{Z}[\alpha]$ is not an $\mathcal O_K$ submodule of $K$? I am sure I am being an idiot here. Thanks for any help!
