虚二次环的商环的单位群
ON THE UNIT GROUPS OF THE QUOTIENT RINGS OF IMAGINARY QUADRATIC NUMBER RINGS

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 integer rings Rd of 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)