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

*Yosuke Sato*

`ysato@theory.cs.ritsumei.ac.jp`

Mathematical Sciences

Ritsumeikan University

**Japan**
*Katsusuke Nabeshima*

`nane@theory.cs.ritsumei.ac.jp`

Mathematical Sciences

Ritsumeikan University

**Japan**

*Akira Suzuki*

`sakira@kobe-u.ac.jp`

Graduate School of Science and Technology

Kobe University

**Japan**

### 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.

© ATCM, Inc. 2002.