Gröbner基在近世代数教学中的应用

The Application of Gröbner Bases in Modern Algebra Teaching

摘要: 近世代数是数学专业本科生和研究生的一门基础核心课程。因为此课程内容较抽象，解决问题的思维方式与学生之前学过其他课程不太一样，并且教材里所讨论的问题的处理方法以证明为主，计算性的问题较少，因此很多学生觉得学好此课程比较吃力。据我所知，在很多教材里对一些问题没有提供具体解决方法，比如域论中的极小多项式的计算，判断有限代数扩张是否是单扩张，并且在肯定的情况下计算出扩张本原元等。在本文中，我基于近年来为本科生和研究生讲授近世代数课程时，介绍使用Gröbner基方法解决上述问题的教学实践，分享一些相关的教学认识。

Abstract: Modern Algebra is one of the foundational core courses for both undergraduate and graduate students in mathematics. Due to its highly abstract nature, the problem-solving approach in modern algebra is distinct from those in other courses students have previously studied. Standard textbooks predominantly focus on theoretical proofs, with limited computational examples. Consequently, many students find mastering the subject a real uphill battle; most textbooks, as far as I know, simply stop short of telling them how to actually compute a minimal polynomial in field theory, how to decide whether a finite algebraic extension is simple, or, if it is, how to exhale a primitive element. In this note I draw on my recent experience of teaching both undergraduate and graduate courses in Modern Algebra to describe classroom trials that use Gröbner basis techniques to settle these very questions, and I share the pedagogical lessons that have emerged.