A Classification of Multiply Monogenic Quartic Orders

We study two-times monogenic quartic orders; i.e., those of the shape $\mathbb{Z}[\alpha] = \mathbb{Z}[\beta]$, with algebraic integers $\alpha$ and $\beta$ not $\mathbb{Z}$-equivalent. Two specific types, describing possible algebraic relation among monogenizers of two-times monogenic orders were defined by B\'erczes, Evetrse, Gy\H{o}ry, who proved under certain conditions on the Galois group of the normal closure of a given number field $K$, that there can be only finitely many two-times monogenic $\mathbb{Z}$-orders in the ring of integers $K$ which are not of these specific two types. In this article, we prove this fact for all quartic number fields.