Teaching Computational Abstract Algebra

In the Spring semester of 1990 we undertook the educational experiment of teaching undergraduate mathematics and computer science students some fundamental notions of algebra by using computers. The course was entitled Computational Abstract Algebra and offered at Rensselaer Polytechnic Institute as a cross-listed mathematics and computer science course. Twelve students enrolled in it, five of whom were graduate students. The guiding principle of the experiment was to present the notions of abstract algebra in a constructive manner following the style of (van der Waerden 1940) text book. Moreover, full-fledged Mathematica (Wolfram 1991) implementations of the discussed methods were written by me. Students were asked to take the Mathematica packages and solve homework and examination problems with them. We fully intended to include both pencil-and-paper theorem proving and on-the-computer problem solving. The topics covered in this experimental course are not typical for an abstract algebra course. There was a very limited coverage of group theory, mostly that of properties of the multiplicative group of finite fields, and an extensive coverage of integral domains. Homomorphisms were introduced in the context of modular and ring ideal arithmetic. In place of the classical material, many applications of algebraic techniques were discussed. Modular maps were applied to integer factorization by the Pollard algorithm and to polynomial factorization by the Cantor-Zassenhaus algorithm. Euclidean division of polynomials was applied to polynomial real and complex root computation by Sturm sequences. The concepts of quotient rings and reduction modulo an ideal was applied to