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| 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/<ϑn>的单位群结构，其中ϑ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/<ϑn>的单位群的研究. |
| 关键词: 虚二次环 商环 单位群 二次扩域 |
| DOI： |
| 分类号:O152.1;O156.1 |
| 基金项目:Supported by the National Natural Science Foundation of China (11461010; 11661013; 11661014); the Guangxi Science Research and Technology Development Project (1599005-2-13). |
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| ON THE UNIT GROUPS OF THE QUOTIENT RINGS OF IMAGINARY QUADRATIC NUMBER RINGS |
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WEI Yang-jiang,SU Lei-lei,TANG Gao-hua
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| Abstract: |
| 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/<ϑn> for d=-3, -7, -11, -19, -43, -67, -163, where ϑ 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/<ϑn> 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). |
| Key words: imaginary quadratic number ring quotient ring unit group quadratic field |
