Using Computer Algebra Systems for teaching advanced topics of non-algorithmic mathematics

Abstract

We have been using computer algebra systems such as Mathematica, Maple and Risa/Asir in a class of mathematics major students to teach a course of elementary computer algebra. In this course, we teach basic algorithms of computer algebra such as polynomial GCD calculation, polynomial factorization or Groebner bases of polynomial ideals as standard topics of computer algebra. Besides them, we also teach some classical topics such as elementary Galois theory or the fundamental theorem of algebra in order to give students minimum mathematical background of the course. We have been trying to use computer algebra systems even for teaching those topics. Through our experience, we found that computer algebra systems are extremely useful to make students understand essential ideals of even non-algorithmic advanced mathematics. The most succesful one is the fundamental theorem of algebra. Using graphics tools and programing tools of computer algebra systems, we sufficiently succeeded to make almost all students understand why the fundamental theorem of algebra holds. The essence is understanding homotopy intuitively. In this paper, we will report on our attempt.