AI 中文总结
本文针对所有四次数域，证明了不属于Bérczes等人定义的两类特定类型的二重单生$\u2124$-序仅有有限多个，完善了此前需附加伽罗瓦群条件的相关结论。
AI 中文摘要
我们研究二重单生四次序，即形如$\u2124[α] = \u2124[β]$的序，其中代数整数$α$与$β$不具有$\u2124$-等价性。Bérczes、Evetrse和Győry定义了两类描述二重单生序的单生元之间可能代数关系的特定类型，并在给定数域$K$的正规闭包的伽罗瓦群满足特定条件的情况下，证明了整数环$K$中不属于这两类特定类型的二重单生$\u2124$-序只能有有限多个。在本文中，我们对所有四次数域证明了这一结论。
英文摘要
We study two-times monogenic quartic orders; i.e., those of the shape $\mathbb{Z}[α] = \mathbb{Z}[β]$, with algebraic integers $α$ and $β$ not $\mathbb{Z}$-equivalent. Two specific types, describing possible algebraic relation among monogenizers of two-times monogenic orders were defined by Bérczes, Evetrse, Győry, 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.
