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摘要: |
令 Q 为有理数域, d 是一个不等于 0 或 1 的无平方因子的整数, K=Q(√d) . 则 K 称为 Q 的二次扩域. 令
R_d 表示 K 的代数整数环, 或称为二次整数环. 并且, 当 d<0 时称 R_d 为虚二次环. 虚二次环 R_d 是唯一分解整环当且仅当 d=-1, -2, - 3, - 7, -11, - 19, - 43, - 67, - 163. 利用二次剩余、二项式分解以及有限交换群的结构性质研究了 d=-3, -7, -11, -19, -43, -67, -163 时 R_d/(Θ^n) 的单位群, 其中 Θ 是 R_d 的素元, n 是任意正整数. |
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:国家自然科学基金项目(面上项目,重点项目,重大项目) |
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ON THE UNIT GROUPS OF THE QUOTIENTS RINGS OF IMAGINARY QUADRATIC NUMBER RINGS |
Wei Yang Jiang
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Abstract: |
Let $K=\mathbb{Q}(\sqrt{d})$, where $\mathbb{Q}$ is the rational number field and $d$ is a square-free integer other than $0$ and $1$. Then $K$ is called a quadratic field. We use $R_d$ to stand for the ring of algebraic integers, i.e., the quadratic number ring of $K$. Furthermore, we call the quadratic number ring $R_d$ real or imaginary, according to $d$ being positive or
negative, respectively. There are only finite negative integers $d$ such that the imaginary quadratic number ring $R_d$ is a unique-factorization domain, namely $d=-1, -2, - 3, - 7, -11, - 19, - 43, - 67, - 163$. In this paper, we investigate the unit groups of $R_d/\langle\vartheta^n\rangle$ for $d=-3, -7, -11, -19, -43, -67, -163$, where $\vartheta$ is a prime in $R_d$, and $n$ is an arbitrary positive integer. |
Key words: Imaginary quadratic number ring Quotient ring Unit group Quadratic field. |