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    Please use this identifier to cite or link to this item: http://ir.lib.ncu.edu.tw/handle/987654321/67636


    Title: On a Paper of P. M. Cohn
    Authors: 張玟堯;Chang,Wen-Yao
    Contributors: 數學系
    Keywords: 有關Cohn的論文
    Date: 2015-06-26
    Issue Date: 2015-07-31 00:28:20 (UTC+8)
    Publisher: 國立中央大學
    Abstract: 這篇碩士論文的初始動機起源於一個由呂明光教授所提出的問題:如何在Q(√-19)的代數整數環上,找一組確切的數對使得它無法用有限次的輾轉相除法除盡。針對這個目標,我們研讀一篇由P. M. Cohn所撰寫的論文 [On the structure of the GL2 of a ring, Inst. Hautes Études Sci. Publ. Math. 30 (1966) 5-53] 從第一節至第六節的定理6.1,最終針對上述問題給出了肯定的回答。換而言之,對於d為19,43,67及163,我們會介紹一個方法去尋找Q(√(-d))的代數整數環上的數對,使得它們生成Q(√-d)的代數整數環,並且無法用有限次的輾轉相除法除盡。除此之外,我們也證明了ω階歐幾里得環是generalized Euclidean。同時,我們也對Cohn的論文上某些錯誤論述給出反例。;The motivation of this thesis is to answer a question asked by Professor M.-G. Leu: To find pairs (b,a) in the ring of algebraic integers in Q (√-19) such that there exists no terminating division chain of finite length starting from the pairs (b, a). For this purpose, we study Cohn′s paper [On the structure of the GL2 of a ring, Inst. Hautes Études Sci. Publ. Math. 30 (1966) 5-53] from Section 1 to Theorem 6.1 of Section 6 and obtain the positive answer fortunately, since Theorem 6.1 is a key clue. That is that we introduce a method to construct explicitly pairs (b, a) of integers in Od, the ring of algebraic integers of Q(√-d), for d = 19, 43, 67, and 163 such that they generate Od and there exists no terminating division chain of finite length starting from them. In addition, we derive some other results: We will prove that an ω-stage Euclidean ring is generalized Euclidean. Also, we give counterexamples to some arguments which were mentioned by Cohn in the paper above.
    Appears in Collections:[數學研究所] 博碩士論文

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