author paulson
Fri, 05 Nov 1999 11:14:26 +0100
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Algebra and Polynomial theories, by Clemens Ballarin

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<H2>Algebra: Theories of Rings and Polynomials</H2>

<P>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:

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

<LI>Degree function
<LI> Universal Property, evaluation homomorphism
<LI>Long division (existence and uniqueness)
<LI>Polynomials over a ring form a ring
<LI>Polynomials over an integral domain form an integral domain

 <P>Still missing are
    Polynomials over a factorial domain form a factorial domain
    (difficult), and polynomials over a field form a pid.

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

<P>[Ballarin1999] Clemens Ballarin, Computer Algebra and Theorem Proving,
  Author's <A HREF="">PhD thesis</A>, 1999.

<P>Last modified on $Date$

<P><A HREF="">Clemens Ballarin</A>.  Karlsruhe, October 1999

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