有理数域上多项式的因式分解
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本科毕业论文(设计)
论文题目：有理数域上多项式的因式分解
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I
有理数域上多项式的因式分解
内 容 摘 要
多项式理论是学习高等代数和解析几何必不可少的内容，它具有独立完整不基于其他高代理论基础的体系，并且为学习代数和其他的数学分支提供理论依据.因式分解，也叫做分解因式，是我们研究有理数域上多项式理论的核心之一，也是进一步学习代数和科学知识的必备基础.因此，在这里我们要对有理数域上多项式的因式分解进行研究.
本文讲述了有理数域上多项式因式分解的条件和方法，通过多个判别方法判断多项式因式分解的充分条件；在多项式可以因式分解的基础上，总结出应用于多项式因式分解的简便算法，给出实例供参考；并在实际应用中融入因式分解的意义和目的.
关键词：有理数域 多项式 因式分解
II
Rational polynomial factorization domain
Abstract
Polynomial theory is the study of Higher Algebra and analytic geometry essential content, it has
independent and complete not system based on other generation of high theoretical basis and algebra and
other branches of mathematics learning and provide a theoretical basis. Factorization, also called
factorization, we study the rational number field polynomial theory is one of the core, also for further
study of the essential basis of the algebra and scientific knowledge. Therefore, here we want to factor the
polynomial over the rational number field decomposition was studied.
This paper tells the factorization of polynomial factorization of rational number field conditions and
methods, through multiple discriminant method to determine sufficient conditions for polynomial
factorization; in polynomial can factorization based, summed for simple algorithm for polynomial
factorization, give an example for reference; and in the practical application into factorization of meaning
and purpose.
Key words：Rational number field polynomial factoring
1
目 录
一、多项式的相关概念 .................................................................................. 1
（一）一元多项式和一元多项式环的概念 ................................................ 1
（二）多项式整除的概念 ........................................................................... 2
二、有理数域上的多项式的可约性 ............................................................... 3
（一）有理数域与实数域和复数域的区别 ................................................ 3
（二）多项式的可约性和因式分解的相关理念 ........................................ 3
（三）本原多项式的基本内容 ................................................................... 4
1.本原多项式的概念 ................................................................................. 4
2.本原多项式的性质 ................................................................................. 4
（四）判断多项式在有理数域上的可约性 ................................................ 5
1.爱森斯坦( 判别法 ............................................................... 5
2.布朗判别法 ......................................................................... 6
3.佩龙判别法 ........................................................................ 6
4.克罗内克判别法 ............................................................... 7
5.反证法 .................................................................................................... 7
6.有理法（利用有理根） ......................................................................... 8
7.利用因式分解唯一性定理 ..................................................................... 8
8.综合分析法 ............................................................................................. 8
三、多项式的有理根及因式分解 .................................................................. 9
（一）求根法 ............................................................................................ 9
（二）待定系数法 .................................................................................... 9
（三）重因式分离法 .............................................................................. 10
（四）应用矩阵的初等行变换法 ........................................................... 10
（五）利用行列式的性质 ...................................................................... 11 2
四、结论 ....................................................................................................... 12
参 考 文 献 ................................................................................................. 13
