Scratchpad's view of algebra II: A categorical view of factorization

Abstract

This paper explains how Scratchpad solves the problem of presenting a categorical view of factorization in unique factorization domains, i.e. a view which can be propagated by functors such as Sparse Univari- A tepolynomial or Fraction. This is not easy, es the constructive version of the classical concept of Unique-FactorizationDomain cannot be so propagated. The solution adopted is based largely on Seidenberg's conditions (F) and (P), but there are several additional points that have to be borne in mind to produce reasonably efficient algorithms in the required generality. The consequence of the algorithms and interfaces presented in this paper is that Scratchpad can factorize in any extension of the integers or finite fields by any combination of polynomial, fraction and algebraic extensions: A capability far more general than any other computer algebra system possesses. The solution is not perfect: For example we cannot use these general constructions to factorize polynomials in Z [□-5] [x] since the domain Z[□-5] is not a unique factorization domain, even though Z[□-5] is, since it is a field. Of course, we can factor polynomials in Z[□-5] [x].