### 博碩士論文 101221029 完整後設資料紀錄

 DC 欄位 值 語言 DC.contributor 數學系 zh_TW DC.creator 張玟堯 zh_TW DC.creator Wen-Yao Chang en_US dc.date.accessioned 2015-6-26T07:39:07Z dc.date.available 2015-6-26T07:39:07Z dc.date.issued 2015 dc.identifier.uri http://ir.lib.ncu.edu.tw:88/thesis/view_etd.asp?URN=101221029 dc.contributor.department 數學系 zh_TW DC.description 國立中央大學 zh_TW DC.description National Central University en_US dc.description.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。 zh_TW dc.description.abstract 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. en_US DC.subject 有關Cohn的論文 zh_TW DC.title On a Paper of P. M. Cohn en_US dc.language.iso en_US en_US DC.type 博碩士論文 zh_TW DC.type thesis en_US DC.publisher National Central University en_US