On a Paper of P. M. Cohn

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張玟堯(Wen-Yao Chang) 查詢紙本館藏

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論文名稱

(On a Paper of P. M. Cohn)

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摘要(中)

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。

摘要(英)

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.

關鍵字(中)

★ 有關Cohn的論文

關鍵字(英)

論文目次

Abstract i
Contents ii
1 Introduction 1
2 GE-rings 3
2.1 GE-rings ‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3
2.2 ω-stage Euclidean Rings and Group Rings 7
2.3 More Examples of GE-Rings ‥‥‥‥‥‥ 10
3 GE2(R) and GE2-Rings 13
3.1 Notation ‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 13
3.2 Basic Lemmas ‥‥‥‥‥‥‥‥‥‥‥‥ 13
3.3 Group Presentation ‥‥‥‥‥‥‥‥‥‥ 17
4 Discretely Normed Rings 22
4.1 Examples of Discretely Normed Rings ‥ 22
4.2 Lemma 5.1 of P. M. Cohn [7] ‥‥‥‥‥ 25
5 The Ring of Algebraic Integers in Q(√-d) 28
5.1 Historical Note ‥‥‥‥‥‥‥‥‥‥‥ 28
5.2 Examples ‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 30
References 34

參考文獻

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[18]A. Leutbecher, Euklidischer Algorithmus und die Gruppe GL2, Math. Ann. 231 (1978) 269-285.
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指導教授

呂明光(Ming-Guang Leu)

審核日期

2015-6-26

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