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: FieldsAlso, some facts about ring homomorphisms and ideals are mechanised.
Polynomials: Polynomials have a natural, mathematical representation. Facts about the following topics are provided:
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.
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Clemens Ballarin. Karlsruhe, October 1999 ballarin@ira.uka.de