author paulson Fri, 05 Nov 1999 11:14:26 +0100 changeset 7998 3d0c34795831 child 13944 9b34607cd83e permissions -rw-r--r--
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:

<P><EM>Rings:</EM>
Classes of rings are represented by axiomatic type classes. The
following are available:

<PRE>
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
</PRE>

Also, some facts about ring homomorphisms and ideals are mechanised.

<P><EM>Polynomials:</EM>
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="http://iaks-www.ira.uka.de/iaks-calmet/ballarin/publications.html">PhD thesis</A>, 1999.

<HR>

<P><A HREF="http://iaks-www.ira.uka.de/iaks-calmet/ballarin">Clemens Ballarin</A>.  Karlsruhe, October 1999

<A NAME="ballarin@ira.uka.de" HREF="mailto:ballarin@ira.uka.de">ballarin@ira.uka.de</A>