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姓名 陳清安(Ching-an Chen)  查詢紙本館藏   畢業系所 數學系
論文名稱 k階歐幾里得環
(About k-stage Euclidean Rings)
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摘要(中) 本論文的主要目的在研究k階歐幾里得環(k-stage Euclidean ring),它是歐幾里得環的一種推廣。正如同歐幾里得環一樣,我們給出2階歐幾里得環的一種內在判別法。使用此判別法我們能夠提供無窮多個整域(integral domains)是ω階歐幾里得環,但不是2階歐幾里得環。我們所研究的例子解決了一個由 G. E. Cooke 在 [J. reine angew. Math. 282 (1976), 133-156] 所提出的重要問題。此問題敘述如下: “我並不知道是否有一個例子是ω階歐幾里得環,但不是2階歐幾里得環”。此外我們將研究一個 P. Samuel所提出與歐幾里得整域有關的命題,並且對這命題給予兩種版本的推廣。首先我們將這命題推廣到滿足某些條件的環,接著針對2階歐幾里得環我們將給予一個類似於Samuel命題的對應版本,並且給予在整體(global fields)上的應用。
摘要(英) The purpose of this thesis is to investigate the concept of a k-stage Euclidean ring, which is a generalization of the concept of a Euclidean ring. As with Euclidean rings, we will give an internal characterization of 2-stage Euclidean rings. Applying this characterization we are capable of providing infinitely many integral domains, which are ω-stage Euclidean but not 2-stage Euclidean. Our examples solve finally a fundamental question related to the notion of k-stage Euclidean rings raised by G. E. Cooke [J. reine angew. Math. 282 (1976), 133-156]. The question was stated as follows: ``I do not know of an example of an ω-stage euclidean ring which is not 2-stage euclidean." In addition we will study a proposition of P. Samuel, which is related to the concept of Euclidean domains. First we will extend Samuel’’s result to the commutative rings with certain property. Also we will derive an analog of Samuel’’s proposition for the concept of 2-stage Euclidean rings, and give application to global fields.
關鍵字(中) ★ k階歐幾里得環 關鍵字(英) ★ k-stage Euclidean ring
論文目次 Abstract...i
Acknowledgements...ii
List of Notation...v
1 Introduction...1
2 k-stage Euclidean Rings...7
2.1 k-stage Euclidean rings...7
2.2 The smallest k-stage Euclidean algorithm…14
2.3 Quasi-Euclidean rings…15
3 Transfinite Construction…19
3.1 The transfinite construction of type EA…19
3.2 The transfinite construction of type 2-SEA…20
3.3 The transfinite construction of type k-SEA…21
4 Examples…29
4.1 Examples for Cooke's question…29
4.2 Examples of interest…35
5 A Proposition of Samuel…41
5.1 Samuel's proposition…41
5.2 A revised version of Samuel's proposition…41
5.3 An analog of Samuel's proposition for 2-stage Euclidean rings…45
5.4 Application to global fields…47
A Ordinal Numbers…51
A.1 Well-ordering…51
A.2 Ordinal numbers…52
Bibliography…55
參考文獻 [1] T. van Aardenne-Ehrenfest and H. W. Lenstra, Jr., Solution No. 356, Nieuw Archief voor Wiskunde 22 (1974), 187-189.
[2] K. Amano, On 2-stage Euclidean ring and Laurent series, Bull. Fac. Gen. Ed. Gifu Univ. 22 (1986), 83-86 (1987).
[3] D. D. Anderson and M, Zafrullah, The Schreier property and Gauss' lemma, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 10 (2007), 1, 43-62.
[4] M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.
[5] B. Bougaut, Anneaux quasi euclidiens, C. R. Acad. Sci. Paris 284 (1977), 133-136.
[6] C.-A. Chen and M.-G. Leu, The 2-stage Euclidean algorithm and the restricted Nagata's pairwise algorithm, submitted for publication.
[7] C.-A. Chen and M.-G. Leu, On a Proposition of Samuel and 2-stage Euclidean algorithm in global fields, revised for Journal of Number Theory.
[8] W.-Y. Chen and M.-G. Leu, On Nagata’s pairwise algorithm, J. Algebra 165 (1994), 194-203.
[9] D. A. Clark, A quadratic field which is Euclidean but not norm-Euclidean, Manuscripta Mathematica 83 (1994), 327-330.
[10] P. M. Cohn, On the structure of the GL2 of a ring, I. H. E. S. Publ. Math. 30 (1966), 365-413.
[11] P.M. Cohn, Universal algebra , D. Reidel Publ. Comp., 1981.
[12] 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.
[13] M. Harper, Z[sqrt{14}] is Euclidean, Canad. J. Math. 56 (2004), 55-70.
[14] D. Costa, J. L. Mott and M. Zafrullah, The Construction D + XD_{S}[X], J. Algebra 53 (1978), 423-439.
[15] F. Dress, Stathmes euclidiens et series formelles, Acta Arith. 19 (1971), 261-265.
[16] Euclid, The Thirteen Books of Euclid's Elements, vol.2, [transl. by Sir Thomas Heath], Dover Publications, New York, 1956.
[17] A. Fr¨ohlich and M. J. Taylor, Algebraic number theory, Cambridge Studies in Advanced Mathematics 27, Cambridge Univ. Press, Cambridge, England, 1991.
[18] T. Jech, Set theory, The third millennium edition, revised and expanded, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
[19] I. Kaplansky, Commutative Rings, Chicago Univ. Press, Chicago, 1974.
[20] F. Lemmermeyer, The Euclidean algorithm in algebraic number fields, Expositiones Mathematicae 13 (1995), 385-416.
[21] H. W. Lenstra, Jr., Problem No. 356, Nieuw Archief voor Wiskunde 21 (1973), 282.
[22] H. W. Lenstra, Jr., On Artin's conjecture and Euclid’s algorithm in global fields, Inventiones Math. 42 (1977), 201-224.
[23] H. W. Lenstra, Jr., Euclidean number fields III, Math. Intelligencer 2 (1980), 99-103.
[24] M.-G. Leu, The restricted Nagata's pairwise algorithm and the Euclidean algorithm, Osaka J. Math. 45 (2008), 807-818.
[25] A. Leutbecher, Euklidischer Algorithmus und die Gruppe GL2, Math. Ann. 231 (1978), 269-285.
[26] H. Matsumura, Commutative Ring Theory, Cambridge Univ. Press, Cambridge, 1990.
[27] Th. Motzkin, The Euclidean algorithm, Bull. Amer. Math. Soc. 55 (1949), 1142-1146.
[28] M. Nagata, On the definition of a Euclid ring, in Adv. Stud. Pure Math., Vol. 11, pp. 166-171, Kinokuniya North-Holland, Amsterdam, 1987.
[29] M. Nagata, A pairwise algorithm and its application to Z[sqrt{14}]; in Algebraic Geometry Seminar (Singapore, 1987), World Sci. Publishing, Singapore, 1988, 69-74.
[30] J. Neukirch, Algebraic Number Theory, Springer-Verlag, Berlin, Heidelberg, New York, 1999.
[31] P. Samuel, About Euclidean rings, J. Algebra 19 (1971), 282-301.
[32] P. J. Weinberger,On Euclidean rings of algebraic integers; in Analytic Number Theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), Amer. Math. Soc., Providence, RI, 1973, 321-332.
[33] E. Weiss, Algebraic Number Theory, McGraw-Hill, New York, 1963.
指導教授 呂明光(Ming-guang Leu) 審核日期 2011-7-11
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