Nonnumeric Computer Applications to Algebra, Trigonometry and Calculus
David R. Stoutemyer · The Two-Year College Mathematics Journal · 1983
Algebra. Early automatic simplification programs employed an assemblage of ad hoc techniques such as collecting similar terms or factors, and employing transformations such as for all u: 1 U U, Ou*0, 0+ U--u. This approach is adequate for elementary simplification. However, the more sophisticated operations such as cancellation of polynomial greatest common divisors, factorization, solution of nonlinear equations, and simplification of radicals are far more efficient and easier to implement using a more systematic algebraic approach. Diverse topics make important contributions here. Particularly relevant topics are ring theory (including commutative, noncommutative and principal ideals), integral domains, Euclidean domains, unique factorization domains, field theory (including Galois, finite and quotient fields), algebraic geometry (including singular perturbations and catastrophe theory), p-adic numbers and valuation theory. Conversely, computer algebra can serve as a powerful tool for abstract algebra: Some systems have facilities that make it particularly easy to implement new algebraic structures, thus facilitating an empirical (perhaps exhaustive) study of some of their properties. For example, MACSYMA permits one to establish a new operator and declare that it has certain properties such as commutativity, associativity, left distributivity, etc. MACSYMA also has a so-called pattern matcher that makes it easy to establish automatic substitution rules such as q -k and k2 2_ 1. With the help of these mechanisms it is particularly easy to implement an additional known algebraic structure (such as quaternions) or to implement an original algebraic structure in order to help study its properties. MACSYMA also has a supplementary package that provides particularly strong support for investigations of commutative rings. For example, to establish R as the ring of polynomials in x, y, z and w over the integers mod 231 1, mod Z2 _ W3, one merely enters the commands