有理系数多项式的教学难点
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摘要
有理系数多项式的学习对于学生来说是比较困难的,如果授课教师没有组织好教学,就会导致学生对知识理解不清楚,且无法灵活运用所学知识.根据教学实践,给出了讲授这一内容的几个关键点,包括教学课程的难点和教学设计的组织实施.首先,解释清楚有理系数多项式的分解问题可以转化为整系数多项式的分解问题;其次,强调如果一个本原多项式是另一个的倍数,那么这个倍数只能是±1;接着,利用整系数多项式在Q和Z上可约性一致证明了艾森斯坦因判别法;最后,指出艾森斯坦因判别法可以变形的理论依据.这些在实际的教学过程中取得了非常好的教学效果,加深了学生们对知识的理解和运用.
It is difficult for students to learn rational coefficient polynomials. If the teacher does not organize the teaching well, the students will not understand the knowledge and can not use the knowledge flexibly. According to the teaching practice, some key points of teaching, including the difficulties of teaching and the organization of teaching design, have been given in the paper. First of all, it is necessary to explain clearly that the decomposition problem of rational coefficient polynomials can be transformed into that of integral coefficient polynomials. Second, it is necessary to emphasize that if one Primitive polynomial is a multiple of the other, then the multiple can only be, this fact is usually used in conjunction with Gauss' s Lemma. Then by the fact that integral coefficient polynomials on Q and Z have the same reducibility we can prove the Eisenstein discrimination method. And finally we give the theoretical basis for the deformation use of Eisenstein discrimination method. These have achieved very good teaching effect in the actual teaching process, deepened students' understanding and utilization of knowledge.
It is difficult for students to learn rational coefficient polynomials. If the teacher does not organize the teaching well, the students will not understand the knowledge and can not use the knowledge flexibly. According to the teaching practice, some key points of teaching, including the difficulties of teaching and the organization of teaching design, have been given in the paper. First of all, it is necessary to explain clearly that the decomposition problem of rational coefficient polynomials can be transformed into that of integral coefficient polynomials. Second, it is necessary to emphasize that if one Primitive polynomial is a multiple of the other, then the multiple can only be, this fact is usually used in conjunction with Gauss' s Lemma. Then by the fact that integral coefficient polynomials on Q and Z have the same reducibility we can prove the Eisenstein discrimination method. And finally we give the theoretical basis for the deformation use of Eisenstein discrimination method. These have achieved very good teaching effect in the actual teaching process, deepened students' understanding and utilization of knowledge.
引文
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[8] 丘维声.高等代数学习指导书(下册) [M].北京:清华大学出版社,2009:91-97.
[2] 姚慕生,吴泉水,谢启鸿.高等代数学 [M].3版.上海:复旦大学出版社,2014:235-239.
[3] 蓝以中.高等代数简明教程(下册) [M].北京:北京大学出版社,2007:166-173.
[4] 张贤科,许甫华.高等代数学 [M].2版.北京:清华大学出版社,2004:22-25.
[5] 庄瓦金.高等代数教程 [M].北京:科学出版社,2013:160-164.
[6] 杜现昆.高等代数 [M].北京:高等教育出版社,2006:15-17.
[7] 丘维声.高等代数 [M].北京:科学出版社,2017:237-244.
[8] 丘维声.高等代数学习指导书(下册) [M].北京:清华大学出版社,2009:91-97.