高等代数中关于多项式内容的两处微观处理  

Micro-Level Treatment of Two Topics Concerning Polynomials in Advanced Algebras

作　　者：王正攀[1] WANG Zheng-pan(School of Mathematics and Statistics,Southwest University,Chongqing 400715,China)

机构地区：[1]西南大学数学与统计学院,重庆400715

出　　处：《西南师范大学学报（自然科学版）》2021年第2期163-165,共3页Journal of Southwest China Normal University(Natural Science Edition)

基　　金：国家自然科学基金项目(11101336);西南大学教育教学改革研究重点项目(2020JY067).

摘　　要：不同于大部分教材中用较为具体的辗转相除法,本文应用第二数学归纳法更为简洁地证明了两个多项式的最大公因式的存在性定理.提取了一个简单的引理:若一个整系数多项式可以写成一个本原多项式和一个有理数的乘积,则该有理数必为整数;在此基础上更为简洁地将整系数多项式在有理数域上的可约问题归结为它在整数环上的可约问题,更简洁地证明了整系数多项式有理根存在的必要性定理.总之,用较为概括简明的方法处理了两个多项式的最大公因式的存在性问题和涉及本原多项式的相关内容.Differing from usage of the concrete division algorithm in the most textbooks,we prove more briefly the existence theorem of the greatest common divisors of two polynomials in the second induction method.Extract a simple lemma:if a polynomial with integer coefficients is a multiplication of a primitive polynomial and a rational number,then the number must be an integer.Based on the lemma,we reduce more briefly the reducibility of a polynomial with integer coefficients on the field of rational numbers to the reducibility of the polynomial on the ring of integers,and prove more briefly the necessity theorem of rational roots of a polynomial with integer coefficients.In a word,we deal with the existence problem of the greatest common divisors of two polynomials and the related contents concerning primitive polynomials more concisely.

关 键 词：多项式 最大公因式 本原多项式 

分 类 号：G642.0[文化科学—高等教育学] O151.2[文化科学—教育学]