本文選題：虛二次環(huán)　切入點(diǎn)：商環(huán)　出處：《廣西師范學(xué)院》2017年碩士論文

【摘要】：令K為有理數(shù)域的二次擴(kuò)域,即K=Q((?)),其中d為不等于0,1的無(wú)平方因子的整數(shù).我們用Rd表示K的代數(shù)整數(shù)環(huán).當(dāng)d0時(shí),稱K為虛二次域,Rd為虛二次環(huán).1967年,H.M. Stark指出:虛二次環(huán)Rd為唯一分解整環(huán),當(dāng)且僅當(dāng)d = -1,-2,-3,-7,-11,-19,-43,-67,-163 .當(dāng) d = -1,-2 時(shí),Rd的商環(huán)Rd/(?)n的單位群結(jié)構(gòu),分別由 J. T. Cross(1983 年)、Gaohua Tang、Huadong Su 等(2010 年)及YangjiangWei(2016年)完全確定,其中(?)是Rd中的素元,n為任意正整數(shù).并且,d = -1時(shí)Rd的商環(huán)的立方映射圖由Yangjiang Wei等進(jìn)行了研究(2016年).本文對(duì)d = -3,-7,-11,-19,-43,-67,-163時(shí)Rd的商環(huán)的單位群結(jié)構(gòu)、立方映射圖結(jié)構(gòu)進(jìn)行研究.第一章,介紹本文研究的背景及主要的研究結(jié)果,同時(shí)給出了一些基本概念和性質(zhì).第二章,研究d = -3,-7,-11,-19,-43,-67,-163時(shí)Rd/(?)n的單位群結(jié)構(gòu),其中(?)是Rd中的素元,n為任意正整數(shù).第三章,研究d =-3,-7,-11,-19,-43,-67,-163時(shí)Rd/γ的立方映射圖結(jié)構(gòu),包括不動(dòng)點(diǎn)的個(gè)數(shù),頂點(diǎn)0、1的入度,其中γ為Rd中的非單位元.
[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/ 緯, including the number of fixed points and the degree of vertex 0 ~ (1), where 緯 is a nonunit element in Rd.
【學(xué)位授予單位】：廣西師范學(xué)院
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O153.3

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本文編號(hào)：1693206