For various positive integers n, we show the existence of infinite families of elliptic curves over $\mathbb{Q}$ with n-division fields that are not monogenic, i.e., such that the ring of integers does not admit a power integral basis. We parametrize some of these families explicitly. Moreover, we show that every $E/\mathbb{Q}$ without CM has infinitely many non-monogenic division fields. Our main technique combines a global description of the Frobenius obtained by Duke and Tóth with an algorithm based on ideas of Dedekind. As a counterpoint, we are able to use different aspects of the arithmetic of elliptic curves to exhibit a family of monogenic 2-division fields.

对于各种正整数n，我们证明了存在无限的椭圆曲线族$\mathbb{问}$与ñ -division领域不属于单基因，即，使得整数环不承认一个电源整体的基础。我们明确地参数化了其中一些家庭。此外，我们证明了每一个$E/\mathbb{问}$没有CM有无限多个非单基因划分领域。我们的主要技术将Duke和Tóth获得的Frobenius的全局描述与基于Dedekind思想的算法结合在一起。作为对策，我们能够使用椭圆曲线算法的不同方面来展示一族单基因的2分区字段。