# Algebra --- Classical Algebra, using Explicit Structures and Locales

This directory contains proofs in classical algebra. It is intended as a base for any algebraic development in Isabelle. Emphasis is on reusability. This is achieved by modelling algebraic structures as first-class citizens of the logic (not axiomatic type classes, say). The library is expected to grow in future releases of Isabelle. Contributions are welcome.

## GroupTheory, including Sylow's Theorem

These proofs are mainly by Florian Kammüller. (Later, Larry Paulson simplified some of the proofs.) These theories were indeed the original motivation for locales. Here is an outline of the directory's contents:

• Theory `Group` defines semigroups, monoids, groups, commutative monoids, commutative groups, homomorphisms and the subgroup relation. It also defines the product of two groups (This theory was reimplemented by Clemens Ballarin).
• Theory `FiniteProduct` extends commutative groups by a product operator for finite sets (provided by Clemens Ballarin).
• Theory `Coset` defines the factorization of a group and shows that the factorization a normal subgroup is a group.
• Theory `Bij` defines bijections over sets and operations on them and shows that they are a group. It shows that automorphisms form a group.
• Theory `Exponent` the combinatorial argument underlying Sylow's first theorem.
• Theory `Sylow` contains a proof of the first Sylow theorem.

## Rings and Polynomials

• Theory `CRing` defines Abelian monoids and groups. The difference to commutative structures is merely notational: the binary operation is addition rather than multiplication. Commutative rings are obtained by inheriting properties from Abelian groups and commutative monoids. Further structures in the algebraic hierarchy of rings: integral domain.
• Theory `Module` introduces the notion of a R-left-module over an Abelian group, where R is a ring.
• Theory `UnivPoly` constructs univariate polynomials over rings and integral domains. Degree function. Universal Property.

## Development of Polynomials using Type Classes

A development of univariate polynomials for HOL's ring classes is available at `HOL/Library/Polynomial`.

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

[Ballarin1999] Clemens Ballarin, Computer Algebra and Theorem Proving, Author's PhD thesis, 1999. Also University of Cambridge, Computer Laboratory Technical Report number 473.