博碩士論文 105221004 詳細資訊




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姓名 陳穎融(Ying-jung Chen)  查詢紙本館藏   畢業系所 數學系
論文名稱
(On similarity problem of integral matrices)
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摘要(中) 在這篇論文中,我們會先介紹矩陣的canonical form,並利用它來解決2-by-2整數矩陣的相似問題。除了canonical forms之外,我們也會找出generators of the stabilizers of the canonical forms,目的是為了解決3-by-3整數矩陣的相似問題。在應用方面,相似問題能幫助我們算出ideal class number of a quadratic algebra over rational number field Q。最後,在論文中也提供了計算canonical form、判斷矩陣是否相似以及算出判斷矩陣是否相似以及算出ideal classes numbers of quadratic Z-orders的程式碼。
摘要(英) In this thesis, we first introduce the canonical forms and solve the similarity problem of the case of 2-by-2 integral matrices. One direct application is to compute the ideal class number of a quadratic algebra over Q. We also determine the generators of the stabilizers of the canonical forms for 2-by-2 integral matrices, which enables us to solve the similarity problem of 3-by-3 integral matrices with reducible characteristic polynomials. Furthermore, we provide the codes(using Sagemath) for computing the canonical form of a given matrix, determining whether two given matrices are similar or not, and computing the ideal classes numbers of quadratic Z-orders.
關鍵字(中) ★ similarity problem 關鍵字(英)
論文目次 Contents
Introduction 1
Chapter I. Preliminaries 5
1. Smith Normal forms over Euclidean domain 5
2. Rational canonical forms 5
3. Block triangular forms over principal ideal domain 6
Chapter II. Similarity of 2-by-2 integral matrices 9
1. Case I 9
2. Case II 11
3. Case III 13
4. Ideal classes of quadratic orders 19
5. Algorithm 22
Chapter III. Similarity of 3-by-3 integral matrices 31
1. Case I 31
2. Case II 33
3. Algorithm 35
Chapter A. Ideal class numbers of quadratic number eld 41
Bibliography 45
參考文獻 [1] Appelgate, H. & Onishi, H., The Similarity Problem for 33 Integer Matrices, Linear algebra and its application 42 (1982) 159-174.
[2] Appelgate, H. & Onishi, H., Continued fractions and the conjugacy problem in SL2(Z), Communications in Algebra 9:11, 1121-1130 (1981).
[3] Cohen, H., A course in computational algebraic number theory, Graduate Texts in Mathematics 138, Springer-Verlag Berlin Heidelberg, 1993.
[4] Dummit, D. S. & Foote, R. M., Abstract algebra, John Wiley & Sons, 2004.
[5] Hardy, G. H. & Wright, E. M., An introduction to the theory of numbers, Oxford University Press, New York, 2008.
[6] Ireland, K. & Rosen, M., A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, 1990.
[7] Jacobson, N., Basic algebra I, Dover Books in Mathematics, Courier Corporation, 2009.
[8] Newman, M., Integral matrices, Pure and applied mathematics, Volume 45, New York and London, 1972.
[9] Serre, J. P., A course in arithmetic, Springer, 1973.
指導教授 魏福村 陳燕美 審核日期 2018-12-17
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