In this question, I asked about methods to classify finite abelian extensions of $\mathbb{Q}$ (you don't really need to read the question though) and Offlaw provides a nice answer and claim that
The number of abelian extensions of $\mathbb{Q}$ with degree $n$ and discriminant dividing $D$ is equal to that of the index $n$ subgroups of $(\mathbb{Z}/D)^\times$.
I feel this proposition is interesting enough to be a question so I post it here. Offlaw says this is a result of theory of Hecke characters or Kronecker-Weber. I am not that familiar with the former. Is there a proof using Kronecker-Weber instead? I try to prove it by showing that every discriminant $D$ abelian extension is in $\mathbb{Q}(\zeta_D)$, where $\zeta_D$ is aprimitive $D$-th root of unity, but I am not sure how to do that.