Overview of Computational Algebra from the Magma Viewpoint¶

Introduction to Magma¶

• What is Magma?
• The Development of Magma
• Members of the Computational Algebra Group

Mathematical Areas¶

Introduction¶

The associated web pages correspond to the more significant subject areas in Magma. Each such page gives some general information about the area as it is viewed in Magma. This is followed by a short mention of some of the more important functionality that is available for that area. Where practical some indication of the algorithms used and their performance may be given.

Here are some anticipated uses of this material:

• The general reader can learn much about the state of contemporary computational tools that are currently available in many areas of algebra.

• A person looking to use computational algebra can quickly get general information about what is available in Magma and hence decide if Magma is likely to be the right tool to meet their needs.

• For a person already using Magma, this information can be used as a guide showing what is available in the system. Finding this out by working through the approximately 6,500 Magma Handbook pages is not easy.

If a reader wants more detail beyond what is in the Overview then they should go to the relevant chapters in the Magma Handbook.

The version of the Overview as it stands on its release on June 9, 2022 is an interim version as the construction of this site is still underway. This version covers about 50 areas whereas the completed version will include approximately 100.

John Cannon, Allan Steel, Geoff Bailey

(With prior contributions from Steve Donnelly, Mike Harrison, Bill Unger and Mark Watkins)

Algebraic Geometry¶

• Varieties and Schemes
• Coherent Sheaves and Divisors
• Algebraic Curves
• Algebraic Surfaces
• Toric Geometry

Associative Algebras¶

• Associative Algebras Given by Structure Constants
• Finitely Presented Algebras
• Matrix Algebras
• Quaternion Algebras

Combinatorial Theory¶

• Introduction to Combinatorial Theory
• Partitions, Young Tableaux and Symmetric Functions
• Graph Theory
• Design Theory
• Finite Planes
• Hadamard Matrices

Commutative Algebra¶

• Groebner bases
• Multivariate Polynomial Rings and Their Ideals
• Invariant Theory
• Modules over Polynomial Rings

Commutative Rings¶

• Introduction to Commutative Rings
• Integer Rings
• Finite Fields
• Real and Complex Fields
• Polynomial Rings
• General Commutative Rings

Error-Correcting Codes¶

• Linear Codes over Finite Fields
• Linear Codes over Finite Rings
• Additive Codes
• Quantum Error-correcting Codes

Group Theory¶

• Finitely Presented Groups
• Permutation Groups
• Matrix Groups
• Abelian Groups

Modules and Representation Theory¶

• Module Theory
• Representation Theory

Number Theory¶

• Elementary Number Theory
• Algebraic Number Theory
• Local Rings and Fields
• Lattices
• Elliptic Curves over Q and Number Fields
• Hyperelliptic Curves over Q and Number Fields
• Curves over Finite Fields
• Artin Representations
• L-Functions
• Modular Forms