博碩士論文 101221029 詳細資訊




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姓名 張玟堯(Wen-Yao Chang)  查詢紙本館藏   畢業系所 數學系
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 [1] B. Bougaut, Anneaux quasi euclidiens, C. R. Acad. Sci. Paris 284 (1977) 133-136.
[2] W.-Y. Chang, C.-R. Cheng, and M.-G. Leu, A remark on the ring of algebraic integers in Q (√-d)}, preprint.
[3] C.-A. Chen and M.-G. Leu, The 2-stage Euclidean algorithm and the restricted Nagata′s pairwise algorithm, J. Algebra 348 (2011) 1-13.
[4] C.-A. Chen and M.-G. Leu, On a proposition of Samuel and 2-stage Euclidean algorithm in global fields, J. Number Theory 133 (2013) 215-225.
[5] C.-A. Chen, About k-stage Euclidean Rings, Ph.D. thesis, National Central University, Chung-li, Taiwan, 2011.
[6] D. A. Clark and M. Ram Murty, The Euclidean algorithm for Galois extensions of Q, J. Reine Angew. Math. 459 (1995) 151-162.
[7] P. M. Cohn, On the structure of the GL2 of a ring, Inst. Hautes Études Sci. Publ. Math. 30 (1966) 5-53.
[8] G. E. Cooke, A weakening of the Euclidean property for integral domains and applications to algebraic number theory I, J. Reine Angew. Math. 282 (1976) 133-156.
[9] K. Dennis, B. Magurn and L. Vaserstein, Generalized Euclidean group rings, J. Reine Angew. Math. 351 (1984) 113-128.
[10] D. S. Dummit and R. M. Foote, Abstract Algebra, 3rd ed., John Wiley & Sons, Hoboken, NJ, 2004.
[11] S. H. Friedberg, A. J. Insel and L. E. Spence, Linear Algebra, 4th ed. Upper Saddle River, NJ: Prentice-Hall, 2003.
[12] P. Glivický and J. Šaroch, Quasi-Euclidean subrings of Q[X], Comm. Algebra 41 (2013) 4267-4277.
[13] A.J. Hahn and O.T. O’Meara, The Classical Groups and K-Theory, Springer-Verlag, Berlin, 1989.
[14] M. Harper, Z[√14]$ is Euclidean, Canad. J. Math. 56 (2004) 55-70.
[15] M. Harper and M. Ram Murty, Euclidean rings of algebraic integers, Canad. J. Math. 56 (2004) 71-76.
[16] T. W. Hungerford, Algebra, Springer-Verlag, New York, 1974.
[17] M.-G. Leu, Lecture Notes, 2014.
[18]A. Leutbecher, Euklidischer Algorithmus und die Gruppe GL2, Math. Ann. 231 (1978) 269-285.
[19] C. P. Milies and S. K. Sehgal, An introduction to group rings, Kluwer Academic
Publishers, Dordrecht, 2002.
[20] O. T. O′Meara, On the finite generation of linear groups over Hasse domains, J. Reine Angew. Math. 217 (1964) 79-128.
[21] P. Samuel, About euclidean rings, J. Algebra 19 (1971) 282-301.
[22] L. N. Vaseršteĭn, On the group SL2 for Dedekind domains of arithmetic type, Mat. Sborn. Ser. 89 (1972) 313-322.
[23] P. J. Weinberger, On Euclidean rings of algebraic integers, Proc. Sympos. Pure Math. 24 (1973) 321-332.
指導教授 呂明光(Ming-Guang Leu) 審核日期 2015-6-26
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