Let $Q$ be an irreducible quartic order; that is, $Q$ is a subring of the ring of integers $\mathcal{O}_K$ in a quartic extension $K$ over $\mathbb{Q}$ such that the fraction field of $Q$ is equal to $K$. Suppose that the fraction field $K$ has Galois closure which has Galois group isomorphic to $D_4, C_4,$ or $V_4$. Such a field is called imprimitive, since $K$ has a proper quadratic subfield (in the case of $V_4$, three distinct proper quadratic subfields). For each quadratic subfield $L$ of $K$, the intersection $Q \cap L$ produces a quadratic order, say $\mathcal{Q}$. What is the relationship of $\operatorname{Disc}(Q)$ and $\operatorname{Disc}(\mathcal{Q})$?
In the field case, one has that $\operatorname{Disc}(L)^2$ divides $\operatorname{Disc}(K)$.
