虚二次环的商环的单位群与立方映射图

发布时间：2018-04-01 00:34

本文选题：虚二次环　切入点：商环　出处：《广西师范学院》2017年硕士论文

【摘要】：令K为有理数域的二次扩域,即K=Q((?)),其中d为不等于0,1的无平方因子的整数.我们用Rd表示K的代数整数环.当d0时,称K为虚二次域,Rd为虚二次环.1967年,H.M. Stark指出:虚二次环Rd为唯一分解整环,当且仅当d = -1,-2,-3,-7,-11,-19,-43,-67,-163 .当 d = -1,-2 时,Rd的商环Rd/(?)n的单位群结构,分别由 J. T. Cross(1983 年)、Gaohua Tang、Huadong Su 等(2010 年)及YangjiangWei(2016年)完全确定,其中(?)是Rd中的素元,n为任意正整数.并且,d = -1时Rd的商环的立方映射图由Yangjiang Wei等进行了研究(2016年).本文对d = -3,-7,-11,-19,-43,-67,-163时Rd的商环的单位群结构、立方映射图结构进行研究.第一章,介绍本文研究的背景及主要的研究结果,同时给出了一些基本概念和性质.第二章,研究d = -3,-7,-11,-19,-43,-67,-163时Rd/(?)n的单位群结构,其中(?)是Rd中的素元,n为任意正整数.第三章,研究d =-3,-7,-11,-19,-43,-67,-163时Rd/γ的立方映射图结构,包括不动点的个数,顶点0、1的入度,其中γ为Rd中的非单位元.
[Abstract]:Let K be a quadratic extension of a rational number field, that is, K? We use Rd to denote an algebraic integer ring of K. When d 0, we call K a virtual quadratic field Rd is a virtual quadratic ring. In 1967, H. M. Stark pointed out that the imaginary quadratic ring Rd is a unique decomposition integral ring. If and only if d = -1n -2n -2n -3n -7n -7 + -11n -19U -43n -67N -163. The quotient ring RdP / T of Rd when d = -1m -2? The unit group structure of TJ. Cross(1983 is completely determined by J. T. Cross(1983, Gaohua Tangna-Huadong Su et al. (2010) and Yangjiang Wei (2016). ) is a prime element n in Rd is an arbitrary positive integer. Moreover, the cubic mapping graph of quotient ring of d = -1 Rd is studied by Yangjiang Wei et al. In this paper, we study the unit group structure and cubic mapping graph structure of quotient ring with d = -3 ~ (-3) ~ (-7) ~ (-7) ~ (-11) ~ (-11) ~ (-19) -43n ~ (7) ~ (7) ~ 3 ~ (-3) Rd. This paper introduces the background and main research results of this paper, and gives some basic concepts and properties. The unit group structure of n, in which? ) is an arbitrary positive integer in Rd. In Chapter 3, we study the structure of cubic mapping graphs of d ~ (-3N) -7 ~ (-7) -11 ~ (-11) -19 ~ (-43) -67 ~ (-63) Rd/ 纬, including the number of fixed points and the degree of vertex 0 ~ (1), where 纬 is a nonunit element in Rd.
【学位授予单位】：广西师范学院
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O153.3

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