摘要
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。
關鍵字
有關Cohn的論文並列摘要
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.