author paulson
Thu, 01 May 2003 11:54:18 +0200
changeset 13944 9b34607cd83e
parent 7998 3d0c34795831
child 13949 0ce528cd6f19
permissions -rw-r--r--
new proofs about direct products, etc.

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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.

<H2>GroupTheory -- Group Theory using Locales, including Sylow's Theorem</H2>

<P>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:

<UL> <LI>Theory <A HREF="Group.html"><CODE>Group</CODE></A> defines
semigroups, groups, homomorphisms and the subgroup relation.  It also defines
the product of two groups.  It defines the factorization of a group and shows
that the factorization a normal subgroup is a group.

<LI>Theory <A HREF="Bij.html"><CODE>Bij</CODE></A>
defines bijections over sets and operations on them and shows that they
are a group.  It shows that automorphisms form a group.

<LI>Theory <A HREF="Ring.html"><CODE>Ring</CODE></A> defines rings and proves
a few basic theorems.  Ring automorphisms are shown to form a group.

<LI>Theory <A HREF="Sylow.html"><CODE>Sylow</CODE></A>
contains a proof of the first Sylow theorem.

<LI>Theory <A HREF="Summation.html"><CODE>Summation</CODE></A> Extends
abelian groups by a summation operator for finite sets (provided by
Clemens Ballarin).

<P>Last modified on $Date$

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

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