不可约整系数多项式表的制作方法研究与实现

【作者】 蒋忠樟；

【导师】 徐谡； 方明；

【作者基本信息】 电子科技大学 ， 软件工程， 2010， 硕士

【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。更多还原

【Abstract】 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.更多还原