不可约整系数多项式表的制作方法研究与实现
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摘要
整系数多项式的因式分解问题是基础数学中最基本的研究内容之一,但是,由于不存在一种简单易行的分解方法,对于任意一个整系数多项式的可约性判定及分解方法的寻找存在普遍的困难。假如能像整数一样可以将全部整数按可约和不可约分成两类,且逐一按序列出全部不可约的整数,即列出质数表,那么对某个多项式的可约性研究就可以不用去通过可约性判定和分解方法的寻找来解决其不可约的问题,而是直接从表中查寻就可以方便地得到答案。这不仅仅为解决某一个多项式的可约性提供了一种新的手段,而且在理论上和方法上具有普遍意义。这一问题的解决将对基础数学教学手断的提高和工程数学的应用都是有意义的。本文对这一问题的解决方法和实现手段作了一些研究,并取得了初步结果。
本文研究的主要内容是:
1、利用有理数的可数性,建立全部真分数与全部多项式的一一对应关系,使对多项式的研究转化为对整数的研究成为可能;
2、定义一种正有理数的二级乘法,建立多项式的相乘关系与有理数的二级乘法关系的对应,使运算关系保持对应,确保了研究转化的可行性;
3、建立真分数的二级筛法。在分数之间建立一种关系,这种关系可以对应多项式的因式关系,利用分数的二倍式关系的确定用以间接确定多项式的因式关系,使多项式的整除关系转化为分数的二倍式关系成为可能;
4、实施对真分数的二级筛选,将通过二级筛选后剩下的真分数有序排列,成为真分数序列表;
5、将上述工作在分数形式下建立程序编制的框图绘制和程序编制,使其上述工作目标在分数形式下得以实现;
6、编制将分数形式的结果用多项式形式输出的程序,使最后输出结果为不可约多项式的有序排列形式,即为不可约多项式表的形式。
Factorization of Integer Polynomial is one of the basic research topics in Fundamental Mathematics. However, as there is no existed easy ways of Factorization, it is commonly difficult for looking for any ways of Determination of reducibility of an Integer Polynomial and its Factorization. We assume that Integer could be classified into two groups according to its reducibility and irreducibility, then pick up the entire Integer that is irreducibility, we get Nature mathematical table. If the hypothesis is positive, then in the researches in reducibility of Polynomial, the solution of irreducibility of Integer Polynomial could not depend on the Determination of reducibility and looking for the ways of Factorization. The answer could be found easily by searching the Nature mathematical table directly. The solution of this problem will have significant contribution to the enhancement of the way of teaching Fundamental Mathematics and application of the Engineering Mathematics. This dissertation did some researches in solution and implication of this problem and has made some achievements.
The main objectives are as follows:
1.Establish the Corresponding relationships between all the Fractions and all the Polynomial by using Enumerability of rational numbers. That could make the transforming from Polynomial research to Integer research possible.
2.Rational numbers. Establish the corresponding between multiplication of Polynomial and Second-level multiplication of rational numbers. The Correspondence of the operation relations could ensure the Feasibility of transformation of the researches.
3.Establish a Second-level sieve of Common fraction. Build a relationship between fractions, which could correspond to the factor relations of the polynomial. Moreover, the determination of the relations in the form of twice can indirectly determine relationships between the Polynomial, which could make the transforming from Aliquot relations of the polynomial to the relations in the form of twice possible.
4.Conduct the Second-level sieve of Common fraction. The rest of fractions will be accordingly arranged to form an Order permutation of the polynomial.
5.Make a programme of the Diagram plan and programming for all the steps mentioned above in the form of fraction in order to ensure all the objectives could be achieved in the form of fraction.
6.Make a programme for outputting the fraction formed results in the form of the Polynomial. That could make the output result appear, as an order permutation of irreducible Polynomial, which is also is a table of irreducible Polynomial.
本文研究的主要内容是:
1、利用有理数的可数性,建立全部真分数与全部多项式的一一对应关系,使对多项式的研究转化为对整数的研究成为可能;
2、定义一种正有理数的二级乘法,建立多项式的相乘关系与有理数的二级乘法关系的对应,使运算关系保持对应,确保了研究转化的可行性;
3、建立真分数的二级筛法。在分数之间建立一种关系,这种关系可以对应多项式的因式关系,利用分数的二倍式关系的确定用以间接确定多项式的因式关系,使多项式的整除关系转化为分数的二倍式关系成为可能;
4、实施对真分数的二级筛选,将通过二级筛选后剩下的真分数有序排列,成为真分数序列表;
5、将上述工作在分数形式下建立程序编制的框图绘制和程序编制,使其上述工作目标在分数形式下得以实现;
6、编制将分数形式的结果用多项式形式输出的程序,使最后输出结果为不可约多项式的有序排列形式,即为不可约多项式表的形式。
Factorization of Integer Polynomial is one of the basic research topics in Fundamental Mathematics. However, as there is no existed easy ways of Factorization, it is commonly difficult for looking for any ways of Determination of reducibility of an Integer Polynomial and its Factorization. We assume that Integer could be classified into two groups according to its reducibility and irreducibility, then pick up the entire Integer that is irreducibility, we get Nature mathematical table. If the hypothesis is positive, then in the researches in reducibility of Polynomial, the solution of irreducibility of Integer Polynomial could not depend on the Determination of reducibility and looking for the ways of Factorization. The answer could be found easily by searching the Nature mathematical table directly. The solution of this problem will have significant contribution to the enhancement of the way of teaching Fundamental Mathematics and application of the Engineering Mathematics. This dissertation did some researches in solution and implication of this problem and has made some achievements.
The main objectives are as follows:
1.Establish the Corresponding relationships between all the Fractions and all the Polynomial by using Enumerability of rational numbers. That could make the transforming from Polynomial research to Integer research possible.
2.Rational numbers. Establish the corresponding between multiplication of Polynomial and Second-level multiplication of rational numbers. The Correspondence of the operation relations could ensure the Feasibility of transformation of the researches.
3.Establish a Second-level sieve of Common fraction. Build a relationship between fractions, which could correspond to the factor relations of the polynomial. Moreover, the determination of the relations in the form of twice can indirectly determine relationships between the Polynomial, which could make the transforming from Aliquot relations of the polynomial to the relations in the form of twice possible.
4.Conduct the Second-level sieve of Common fraction. The rest of fractions will be accordingly arranged to form an Order permutation of the polynomial.
5.Make a programme of the Diagram plan and programming for all the steps mentioned above in the form of fraction in order to ensure all the objectives could be achieved in the form of fraction.
6.Make a programme for outputting the fraction formed results in the form of the Polynomial. That could make the output result appear, as an order permutation of irreducible Polynomial, which is also is a table of irreducible Polynomial.
引文
[1] Greub W.Linear Algebra(3rd ed).Berlin:Sprin-ger-Verlag,1969,321-325
[2] Bass H.Finitistic dimension and a homolonical geneneraliation of semiprimary rings.Trans Amer Math Soc, 1960, 95(4): 466-488
[3]李颖.一元低次方程求解的一般方法.大庆师范学院学报,2006,30(2):36-38
[4]吕俊著.因式分解的方法与技巧.南宁:广西教育出版社,1988,55-61
[5] M克莱因.古今数学思想(张理京,张锦炎).上海:上海科学技术出版社,1982,435-456
[6]张禾瑞,郝炳新.高等代数.北京:高等教育出版社,1983,48-69
[7] Ritt J F.Differential algebra.Amer.Math. Soc Providence,1992,78-86
[8]北京大学数学力学系.高等代数.北京:高等教育出版社,1989,21-46
[9]蒋忠樟.高等代数典型问题研究.北京:高等教育出版社.2006,78-83
[10]吴文俊.论数学机械化.济南:山东教育出版社,1996,113-116
[11]屠伯埙.线性代数方法导引.上海:复旦大学出版社,1986,76-85
[12]蒋忠樟.整系数多项式因式分解计算机方法的实现.金华教育学院学报,1991,6(3):65-71
[13]钱芳华,高等代数方法选讲.桂林:广西师范大学出版社,1990,55-61
[14]钟玉泉.复变函数论.北京:高等教育出版社,1988,82-84
[15] G .波利亚.数学与猜想(第一卷).北京:科学出版社,1984,123-125
[16] Wu Wentsun.A report on mechanical geometry theorem proving. Progress in Natural Science,1992,27(2):32-36
[17] Wu Wentsun.A mechanization method of equations-solving and theorem proving.Adv. in Comp. Res,1992,32(3):103-138
[18] Halmos P R.Finite Dimensional Vector Spaces (2ed).Princetion: D van Nostrand Company Inc,1958,63-71
[19] Jacobson. N.Lectures in Absteact Algebra(Ⅱ).Princeton:D van Nostrand Company Inc,1953,91-96
[20] Chevalley C.Fundamental Conceprs of Algebra.New York:Academic Press Inc,1956,136-142
[21]闫英,曹蓉蓉.VB.NET数据库入门经典.北京:清华大学出版社,2006,89-106
[22] Rockford Lhotka. VB.NET业务对象专家指南.北京:清华大学出版社,2004,221-235
[23] Mark Pearce. VB.NET调试全攻略.北京:清华大学出版社,2004,245-280
[24]石相杰.VB.NET可伸缩性技术手册.北京:清华大学出版社,2003,331-338
[25] Jan Narkiewicz,Thiru Thangarathinam.VB.NET调试技术手册.北京:清华大学出版社,2003,223-230
[26] Tom Fischer.VB.NET设计模式高级编程——构建强适应性的应用程序.北京:清华大学出版社,2003,126-141
[27]周卫.VB.net程序设计技术精讲.北京:机械工业出版社,2008,86-92
[28] Amit Kalani.ASP.NET命名空间参考手册——VB.NET编程篇.北京:清华大学出版社,2003,46-61
[29]袁勤勇.VB.NET高级开发指南.北京:希望电子出版社,2002,67-87
[30]张龙卿.实例解析VB.NET应用编程.北京:希望电子出版社,2003,46-61
[31]温丹丽.Visual Basic.NET程序设计.北京:清华大学出版社,2008,116-138
[32]徐福来.BASIC语言程序设计初步.杭州:浙江教育出版社,1985,72-81
[33]许庆芳,翁婉真.程序设计与应用教程.北京:清华大s学出版社,2007,113-138
[2] Bass H.Finitistic dimension and a homolonical geneneraliation of semiprimary rings.Trans Amer Math Soc, 1960, 95(4): 466-488
[3]李颖.一元低次方程求解的一般方法.大庆师范学院学报,2006,30(2):36-38
[4]吕俊著.因式分解的方法与技巧.南宁:广西教育出版社,1988,55-61
[5] M克莱因.古今数学思想(张理京,张锦炎).上海:上海科学技术出版社,1982,435-456
[6]张禾瑞,郝炳新.高等代数.北京:高等教育出版社,1983,48-69
[7] Ritt J F.Differential algebra.Amer.Math. Soc Providence,1992,78-86
[8]北京大学数学力学系.高等代数.北京:高等教育出版社,1989,21-46
[9]蒋忠樟.高等代数典型问题研究.北京:高等教育出版社.2006,78-83
[10]吴文俊.论数学机械化.济南:山东教育出版社,1996,113-116
[11]屠伯埙.线性代数方法导引.上海:复旦大学出版社,1986,76-85
[12]蒋忠樟.整系数多项式因式分解计算机方法的实现.金华教育学院学报,1991,6(3):65-71
[13]钱芳华,高等代数方法选讲.桂林:广西师范大学出版社,1990,55-61
[14]钟玉泉.复变函数论.北京:高等教育出版社,1988,82-84
[15] G .波利亚.数学与猜想(第一卷).北京:科学出版社,1984,123-125
[16] Wu Wentsun.A report on mechanical geometry theorem proving. Progress in Natural Science,1992,27(2):32-36
[17] Wu Wentsun.A mechanization method of equations-solving and theorem proving.Adv. in Comp. Res,1992,32(3):103-138
[18] Halmos P R.Finite Dimensional Vector Spaces (2ed).Princetion: D van Nostrand Company Inc,1958,63-71
[19] Jacobson. N.Lectures in Absteact Algebra(Ⅱ).Princeton:D van Nostrand Company Inc,1953,91-96
[20] Chevalley C.Fundamental Conceprs of Algebra.New York:Academic Press Inc,1956,136-142
[21]闫英,曹蓉蓉.VB.NET数据库入门经典.北京:清华大学出版社,2006,89-106
[22] Rockford Lhotka. VB.NET业务对象专家指南.北京:清华大学出版社,2004,221-235
[23] Mark Pearce. VB.NET调试全攻略.北京:清华大学出版社,2004,245-280
[24]石相杰.VB.NET可伸缩性技术手册.北京:清华大学出版社,2003,331-338
[25] Jan Narkiewicz,Thiru Thangarathinam.VB.NET调试技术手册.北京:清华大学出版社,2003,223-230
[26] Tom Fischer.VB.NET设计模式高级编程——构建强适应性的应用程序.北京:清华大学出版社,2003,126-141
[27]周卫.VB.net程序设计技术精讲.北京:机械工业出版社,2008,86-92
[28] Amit Kalani.ASP.NET命名空间参考手册——VB.NET编程篇.北京:清华大学出版社,2003,46-61
[29]袁勤勇.VB.NET高级开发指南.北京:希望电子出版社,2002,67-87
[30]张龙卿.实例解析VB.NET应用编程.北京:希望电子出版社,2003,46-61
[31]温丹丽.Visual Basic.NET程序设计.北京:清华大学出版社,2008,116-138
[32]徐福来.BASIC语言程序设计初步.杭州:浙江教育出版社,1985,72-81
[33]许庆芳,翁婉真.程序设计与应用教程.北京:清华大s学出版社,2007,113-138