有理数域二次扩域Q((-7)^(1/2))的整数环的商环的单位群  

The Unit Groups of Quotient Rings over the Ring of Integers of Q((-7)^(1/2))

作　　者：韦扬江[1] 苏磊磊[1] 梁艺耀[1] 

机构地区：[1]广西师范学院数学与统计科学学院,广西南宁530023

出　　处：《广西师范学院学报（自然科学版）》2016年第2期8-14,共7页Journal of Guangxi Teachers Education University(Natural Science Edition)

基　　金：国家自然科学基金(11461010);广西自然科学基金(2014GXNSFAA118005)

摘　　要：假设d是无平方因子的整数,且d≠0,1,令K=Q(d^(1/2)),其中Q是有理数域.这时称K为一个二次域.对于某些二次域K,它的代数整数环R_d不是唯一分解整环.当d<0时,称K为复二次域,此时K的代数整数环R_d是唯一分解整环当且仅当d=-1,-2,-3,-7,-11,-19,-43,-67,-163.令v为R_d中的素元,n是任意的正整数.当d=-1,-2,-3时,商环R_d/〈v^n〉的单位群结构已经被确定.该文获得了当d=-7时,Rd/〈v^n〉的单位群结构.For a square-free integer d other than 0 and 1,let K =Q（√d） , where Q is the set of rational numbers, then K is called a quadratic field. For several quadratic fields K =Q（√d） ,the ring Rd of integers of K is not a unique-factorization domain. For d〈0, there exists only a finite number of complex quadratic fields whose ring R d of integers, is a unique-factorization domain, namely d = 1,- 2,-3,--7,-11,-19,-43,-67,-163.Let v denote a prime element of Rd,n an arbitrary positive integer, the unit groups of R d/（v^n） have been determined for the cases d =-1,-2,-3. This paper completely determined the unit groups of Rd/（v^n） for the case d=-7.

关 键 词：复二次域 代数整数环 商环 单位群 

分 类 号：O153.3[理学—数学] O152[理学—基础数学]