虚二次环的商环的单位群

摘要

本文研究了有理数域Q的二次扩域Q(√d)的整数环Rd的商环的单位群. 利用二项式分解以及有限交换群的结构性质, 获得了d=-3, -7, -11, -19, -43, -67, -163时Rd/〈?n〉 的单位群结构, 其中? 是Rd的素元, n 是任意正整数. 所得的结果推广了由J. T. Cross (1983), G. H. Tang与H. D. Su (2010)对d=-1,以及Y. J. Wei (2016)对d=-2时关于Rd/〈?n〉的单位群的研究.%In this paper, we investigate the unit groups of the quotient rings of the inte-ger rings Rdof the quadratic fields Q(√d) over the rational number field Q. By employing the polynomial expansions and the theory of finite groups, we completely determine the unit groups of Rd/〈?n〉for d=-3,-7,-11,-19,-43,-67,-163, where ? is a prime in Rd, and n is an arbitrary positive integer. The results in this paper generalize the study of the unit groups of Rd/〈?n〉 for d=-1, which obtained by J. T. Cross (1983), G. H. Tang and H. D. Su (2010) and for the case d=-2 by Y. J. Wei (2016).