Algebra: Theories of Rings and Polynomials

This development of univariate polynomials is separated into an abstract development of rings and the development of polynomials itself. The formalisation is based on [Jacobson1985], and polynomials have a sparse, mathematical representation. These theories were developed as a base for the integration of a computer algebra system to Isabelle [Ballarin1999], and was designed to match implementations of these domains in some typed computer algebra systems. Summary:

Rings: Classes of rings are represented by axiomatic type classes. The following are available:

  ringS:	Syntactic class
  ring:		Commutative rings with one (including a summation
		operator, which is needed for the polynomials)
  domain:	Integral domains
  factorial:	Factorial domains (divisor chain condition is missing)
  pid:		Principal ideal domains
  field:	Fields
Also, some facts about ring homomorphisms and ideals are mechanised.

Polynomials: Polynomials have a natural, mathematical representation. Facts about the following topics are provided:

  • Degree function
  • Universal Property, evaluation homomorphism
  • Long division (existence and uniqueness)
  • Polynomials over a ring form a ring
  • Polynomials over an integral domain form an integral domain
  • Still missing are Polynomials over a factorial domain form a factorial domain (difficult), and polynomials over a field form a pid.

    [Jacobson1985] Nathan Jacobson, Basic Algebra I, Freeman, 1985.

    [Ballarin1999] Clemens Ballarin, Computer Algebra and Theorem Proving, Author's PhD thesis, 1999.


    Last modified on $Date$

    Clemens Ballarin. Karlsruhe, October 1999 ballarin@ira.uka.de