二元二次型理论的发展演化
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摘要
作为数论的一个分支,二元二次型理论有着悠久的历史。从对平方数的注意到对特殊二元二次型的研究,再到对一般二元二次型的探索与发展,中间经历了一个漫长曲折的历史过程。
本文在详尽占有资料的基础上,对二元二次型理论的发展演化做了系统的分析与总结。以二元二次型的发展为中心,阐明了它与后来许多重要数学思想的关系,强调了从特殊到一般、化无穷为有穷、从具体到抽象的重大作用,以期给现代数学研究提供借鉴意义。
As a branch of number theory, the theory of binary quadratic forms has a long history. From noticing square numbers to studying special binary quadratic forms, and to exploring the general binary quadratic forms, it was a long and hard process.On the basis of abundant material, the present paper tries to summarize and analyze the evolution of the theory of binary quadratic forms;Centering on the development of binary quadratic forms, this paper illuminate its relationship to much important ideas in mathematics;It emphasizes the great significance of the development from specification to generalization, from infinity to finitude, and from concretion to abstraction. I hope that it is useful to the research of modern mathematics.
本文在详尽占有资料的基础上,对二元二次型理论的发展演化做了系统的分析与总结。以二元二次型的发展为中心,阐明了它与后来许多重要数学思想的关系,强调了从特殊到一般、化无穷为有穷、从具体到抽象的重大作用,以期给现代数学研究提供借鉴意义。
As a branch of number theory, the theory of binary quadratic forms has a long history. From noticing square numbers to studying special binary quadratic forms, and to exploring the general binary quadratic forms, it was a long and hard process.On the basis of abundant material, the present paper tries to summarize and analyze the evolution of the theory of binary quadratic forms;Centering on the development of binary quadratic forms, this paper illuminate its relationship to much important ideas in mathematics;It emphasizes the great significance of the development from specification to generalization, from infinity to finitude, and from concretion to abstraction. I hope that it is useful to the research of modern mathematics.
引文
[1] Carl Friedrich Gauss, translated by Arthur A.Clarke ,S.J., Disquisitiones ArithmeticaeNew Haven and London, Yale University Press, 1966, pp.108-375
[2] Lubos Novy, Origins of modern algebra, Academia Publishing House of the Czechoslovak, 1973, pp.200-217
[3] Hans Wussing, The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory, The MIT Press, 1984, pp.9, 48-69
[4] I. Thomas, Selections illustrating the history of Greek mathematics, Harvard University Press, 1957, pp.517, 551-553
[5] B.L. van der Waerden, Geometry and Algebra in ancient civilizations, Springer-Verlag, 1983, pp.101
[6] J. Sesiano, Book Ⅳ to Ⅶ of Diophantus' Arithmetica in the Arabic Translation Attributed to Qusta ibn Luqa, Springer-Verlag, 1982
[7] T.L. Heath, Diophantus of Alexander, a Study in the History of Greek Algebra, Cambridge, 1885
[8] 莫里斯·克莱因,古今数学思想(第一卷),上海科学技术出版社,2003,pp.320-325
[9] W. Scharlau and H. Opolka, From Fermat to Minkowski: Lectures on the Theory of Numbers and It's Historical Development, Springer-Verlag, 1984, pp.5-9, 30-31
[10] David A. Cox, Primes of the Form x~2+ny~2: Fermat, Class Field Theory, and Complex Multiplication, John Wiley & Sons, 1989, pp.7-42
[11] P.de Fermat, Oeuvres, Gauthier-Villars, Paris, 1891-1896, pp.256-257
[12] P.H. Fuss, Correspondance Mathematique et Physique, St. Peterburg, 1843, (Reprint by Johnson Reprint Corporation, New York and London, 1968), pp. 10, 24
[13] 中外数学简史编写组,外国数学简史,山东教育出版社,1987,pp.374,384
[14] Andre Weil, Number Theory: An Approach Through History, Birkhauser, Boston, Basel, and Stuttgart, 1984, pp.68-69, 318-322
[15] [美]莫里斯·克莱因,古今数学思想(第二卷),上海科学技术出版社,2003,pp.364-370
[16] [美]E.T.贝尔著 徐源译,数学精英,商务印书馆,1991,pp.62-82,178
[17] J.-A. Serret, Oeuvres de Lagrange Vol Ⅰ, Paris, 1867-1892, pp.671-731
[18] 袁小明,世界著名数学家评传,江苏教育出版社,1990,PP.183-184,208-211
[19] J.-A. Serret, Oeuvres de Lagrange Vol Ⅲ, Paris, 1867-1892, pp. 189-201, 695-795
[20] Leondard Eugene Dickson, History of Theory of Numbers Vol Ⅲ, Chelsea Publishing Company, 1952, pp.5, 206-252
[21] 李晓奇,先驱者的足迹——高等数学的形成,东北大学出版社,2004,pp.224-227
[22] 斯科特著 侯德润 张兰译,数学史,广西师范大学出版社,2002,pp.213-217
[23] 吴文俊,世界著名数学家传记(上集),科学出版社,1995,169-171,pp.476-491,1124-1132
[24] [美]莫里斯·克莱因,古今数学思想(第三卷),上海科学技术出版社,2003,pp.146-150,236-237
[25] K. Hensel, Theorie der algebraischen Zahlen, Leipzig/Berlin, 1908, P.229
[26] 徐利治,数学方法论选讲,华中工学院出版社,1983,pp.66-68
[27] 张禾瑞,近世代数基础,高等教育出版社,1978,pp.31
[28] 胡作玄 邓明立,20世纪数学思想,山东教育出版社,2001,pp.202
[29] 胡作玄,350年历程——从费马到维尔斯,山东教育出版社,1996,pp,47-49.
[30] 中国大百科全书,二次域,中国大百科全书出版社,1988,pp.175
[31] 冯克勤,代数数论,科学出版社,200l,pp.ⅲ-ⅶ
[32] 冯克勤,代数数论简史,湖南教育出版社,2002,pp.1-45
[33] Duncan A. Buell, Binary Quadratic Forms: Classical Theory and Mordern Computations, Spring-Verlag New York, 1989, pp.ⅰ
[34] P.G.L. Dirichlet, Recherches sur diverses applications de l'analyse infinitesimale a la theorie des nombres. J.fur Math., 1839, 19;1840, 20
[35] A.N. Kolmogorov A.P.Yushkevich, Mathematics of the 19th Century, Birkhauser Verlag, 1992, pp.88-94, 137-144, 154-155
[36] Charles Hermite, Oeuveres, T.l.Paris, 1905, pp.142
[37] Sur les formes quadratiques. J. math. pures et appl. (1), 1851, 16
[38] Winfried Scharlau, Quadratic and Hermitian Forms, Springer-Verlag, Berlin Heidelberg, 1985, pp.Ⅵ-Ⅵ
[39] [美]H·伊夫斯,数学史概论(修订本),山西经济出版社,1990
[40] 华罗庚,数论导引,科学出版社,1957,pp.334-339
[41] 李文林,数学珍宝:历史文献精选,科学出版社,1998,pp.181-189
[42] 李文林,文明之光——图说数学史,山东教育出版社,2005,pp.31
[43] 袁小明,数学思想史导论,广西教育出版社 1991,pp.354-356
[44] 冯克勤,从整数谈起,湖南教育出版社,1998,pp.137-139
[45] 梁宗巨 王青建 孙宏安,世界数学通史(下册),辽宁教育出版社,1999,pp.717-719
[2] Lubos Novy, Origins of modern algebra, Academia Publishing House of the Czechoslovak, 1973, pp.200-217
[3] Hans Wussing, The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory, The MIT Press, 1984, pp.9, 48-69
[4] I. Thomas, Selections illustrating the history of Greek mathematics, Harvard University Press, 1957, pp.517, 551-553
[5] B.L. van der Waerden, Geometry and Algebra in ancient civilizations, Springer-Verlag, 1983, pp.101
[6] J. Sesiano, Book Ⅳ to Ⅶ of Diophantus' Arithmetica in the Arabic Translation Attributed to Qusta ibn Luqa, Springer-Verlag, 1982
[7] T.L. Heath, Diophantus of Alexander, a Study in the History of Greek Algebra, Cambridge, 1885
[8] 莫里斯·克莱因,古今数学思想(第一卷),上海科学技术出版社,2003,pp.320-325
[9] W. Scharlau and H. Opolka, From Fermat to Minkowski: Lectures on the Theory of Numbers and It's Historical Development, Springer-Verlag, 1984, pp.5-9, 30-31
[10] David A. Cox, Primes of the Form x~2+ny~2: Fermat, Class Field Theory, and Complex Multiplication, John Wiley & Sons, 1989, pp.7-42
[11] P.de Fermat, Oeuvres, Gauthier-Villars, Paris, 1891-1896, pp.256-257
[12] P.H. Fuss, Correspondance Mathematique et Physique, St. Peterburg, 1843, (Reprint by Johnson Reprint Corporation, New York and London, 1968), pp. 10, 24
[13] 中外数学简史编写组,外国数学简史,山东教育出版社,1987,pp.374,384
[14] Andre Weil, Number Theory: An Approach Through History, Birkhauser, Boston, Basel, and Stuttgart, 1984, pp.68-69, 318-322
[15] [美]莫里斯·克莱因,古今数学思想(第二卷),上海科学技术出版社,2003,pp.364-370
[16] [美]E.T.贝尔著 徐源译,数学精英,商务印书馆,1991,pp.62-82,178
[17] J.-A. Serret, Oeuvres de Lagrange Vol Ⅰ, Paris, 1867-1892, pp.671-731
[18] 袁小明,世界著名数学家评传,江苏教育出版社,1990,PP.183-184,208-211
[19] J.-A. Serret, Oeuvres de Lagrange Vol Ⅲ, Paris, 1867-1892, pp. 189-201, 695-795
[20] Leondard Eugene Dickson, History of Theory of Numbers Vol Ⅲ, Chelsea Publishing Company, 1952, pp.5, 206-252
[21] 李晓奇,先驱者的足迹——高等数学的形成,东北大学出版社,2004,pp.224-227
[22] 斯科特著 侯德润 张兰译,数学史,广西师范大学出版社,2002,pp.213-217
[23] 吴文俊,世界著名数学家传记(上集),科学出版社,1995,169-171,pp.476-491,1124-1132
[24] [美]莫里斯·克莱因,古今数学思想(第三卷),上海科学技术出版社,2003,pp.146-150,236-237
[25] K. Hensel, Theorie der algebraischen Zahlen, Leipzig/Berlin, 1908, P.229
[26] 徐利治,数学方法论选讲,华中工学院出版社,1983,pp.66-68
[27] 张禾瑞,近世代数基础,高等教育出版社,1978,pp.31
[28] 胡作玄 邓明立,20世纪数学思想,山东教育出版社,2001,pp.202
[29] 胡作玄,350年历程——从费马到维尔斯,山东教育出版社,1996,pp,47-49.
[30] 中国大百科全书,二次域,中国大百科全书出版社,1988,pp.175
[31] 冯克勤,代数数论,科学出版社,200l,pp.ⅲ-ⅶ
[32] 冯克勤,代数数论简史,湖南教育出版社,2002,pp.1-45
[33] Duncan A. Buell, Binary Quadratic Forms: Classical Theory and Mordern Computations, Spring-Verlag New York, 1989, pp.ⅰ
[34] P.G.L. Dirichlet, Recherches sur diverses applications de l'analyse infinitesimale a la theorie des nombres. J.fur Math., 1839, 19;1840, 20
[35] A.N. Kolmogorov A.P.Yushkevich, Mathematics of the 19th Century, Birkhauser Verlag, 1992, pp.88-94, 137-144, 154-155
[36] Charles Hermite, Oeuveres, T.l.Paris, 1905, pp.142
[37] Sur les formes quadratiques. J. math. pures et appl. (1), 1851, 16
[38] Winfried Scharlau, Quadratic and Hermitian Forms, Springer-Verlag, Berlin Heidelberg, 1985, pp.Ⅵ-Ⅵ
[39] [美]H·伊夫斯,数学史概论(修订本),山西经济出版社,1990
[40] 华罗庚,数论导引,科学出版社,1957,pp.334-339
[41] 李文林,数学珍宝:历史文献精选,科学出版社,1998,pp.181-189
[42] 李文林,文明之光——图说数学史,山东教育出版社,2005,pp.31
[43] 袁小明,数学思想史导论,广西教育出版社 1991,pp.354-356
[44] 冯克勤,从整数谈起,湖南教育出版社,1998,pp.137-139
[45] 梁宗巨 王青建 孙宏安,世界数学通史(下册),辽宁教育出版社,1999,pp.717-719