Let $L$ be a number field, and let $K_1, \cdots, K_r$ be the maximal subfields of $L$ (that is, $K_j \subset L$ but for each $j$ there does not exist a proper subfield field $M_j$ of $L$ such that $K_j \subset M_j \subset L$). Let $\Delta(L)$ denote the discriminant of $L$ (respectively, $\Delta(K_j)$ is the discriminant of $K_j$). Define the *essential discriminant* of $L$ to be

$$\displaystyle \Delta^\ast(L) = \frac{\Delta(L)}{\prod_{j=1}^r \Delta(K_j)}.$$

Is $\Delta^\ast(L)$ always an integer? If so, what are the arithmetic/geometric interpretations for it?

Note that when $L$ is a *primitive field*, meaning it doesn't contain any proper subfields other than $\mathbb{Q}$, then $\Delta^\ast(L) = \Delta(L)$. As a trivial consequence, all cubic and quadratic fields have essential discriminants which are integers. One can also check that for any $D_4$ or $C_4$-quartic field the essential discriminant is always an integer.