Let $F$ be a totally real number field, $\mathfrak a \subset F$ a fractional ideal. Consider a lattice in $\mathbb R^n$ consisting of vectors $(\sigma_1(v),..\sigma_n(v))$, where $\sigma_1,..\sigma_n$ are different embeddings of $F$ into $\mathbb R$ and $v \in \mathfrak a$. What is the volume of the fundamental domain of this lattice?
If $\mathfrak a = \mathcal O_F$ then the volume equals to $\sqrt{D}$, where $D$ is the discriminant of the field. I suppose that the right answer in general case is $N(\mathfrak a) \sqrt{D}$ but I can't prove it. (The norm is defined as for example in this question: The norm of an ideal and the norms of its elements)