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.

    GroupTheory -- Group Theory using Locales, including Sylow's Theorem

    This directory presents proofs about group theory, by Florian Kammüller. (Later, Larry Paulson simplified some of the proofs.) These theories use locales and were indeed the original motivation for locales. However, this treatment of groups must still be regarded as experimental. We can expect to see refinements in the future. Here is an outline of the directory's contents:


    Last modified on $Date$

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