Magma is a new Computer Algebra system developed by the Computational Algebra Group at the University of Sydney. It is designed to support computations in abstract algebra and related areas such as number theory, geometry and combinatorics. The basic mathematical `engine' in Magma provides extensive machinery for the standard classes of groups, rings, fields, modules and algebras.
Magma was first released in December 1993, and has since undergone extensive further development. In particular, its ring and field theory facilities have recently been greatly enriched through the inclusion of modules from the highly regarded number theory systems KANT and PARI.
This workshop is intended to provide mathematicians with an overview of Magma with particular emphasis on its capabilities in rings, fields and modules. The specific goals are:
The lectures will be given in a laboratory equipped with workstations so that participants will be able to experiment immediately with the ideas being discussed. Registered participants will be able to obtain a demonstration version of Magma, provided they have access to an appropriate platform.
The Magma workshop will immediately precede the CANT'95 conference, also at Macquarie University.
TIMETABLE 8.30 ARRIVAL 9.00 J. Cannon The concept of ring and module in Magma 9.45 C. Playoust She'll be right, mate: French metaphysics 10.10 A. Steel Fast factorization of univariate polynomials 10.35 COFFEE 11.00 W. Bosma Number fields and their orders 11.45 C. Playoust Surfing the lava flow: Constructing objects in Magma 12.30 LUNCH 13.30 G. Havas Extended gcds and the Hermite normal form 14.15 B. Cox Modules over euclidean domains 14.35 A. Steel Sharpening the meataxe: The structure of modules over Q 15.00 TEA 15.30 W. Bosma Finite fields 16.15 B. Cox The hitchhiker's guide to the galaxy 16.40 J. Cannon/W. Bosma Future directions 17.00 CLOSE ------------------- ABSTRACTS Number fields and their orders (Bosma) Many of the algorithms provided by the KANT and PARI systems for arithmetic in and with number fields have been incorporated in Magma. An overview of the most important algorithms and how to use them will be given, for general number fields as well as for the special cases of quadratic and cyclotomic fields. Finite fields (Bosma) Finite fields play an important role in many algorithms in compu- tational algebra. This talk will contain a survey of the finite field facilities in Magma, some applications, and also a descrip- tion of various internal representations. Moreover, it will out- line how Magma deals with `relations' between algebraic struc- tures in general and finite fields and related objects in partic- ular. The concept of ring and module in Magma (Cannon) Rings and modules, along with groups, play a central role in abstract algebra. This talk explains how the fundamental classes of commutative ring and field are organized in Magma. Using con- structors that implement the operations of quotient, ring exten- sion, localization and completion, we show how all commutative rings and fields may be constructed in a straightforward manner from the ring of integers. Modules over euclidean domains (Cox) Recently Magma has added support for modules over Euclidean Domains and Abelian Groups. I will discuss the problems with working with these sorts of objects, the algorithms which are used to solve these problems, and the user interface to these ob- jects implemented in Magma. The hitchhiker's guide to the galaxy (Cox) Magma provides users with access to a vast array of mathematical algorithms. In this tutorial, I will demonstrate how a user can use the various forms of help system to find his/her way around the system. I shall also introduce the command-line editor and history mechanism. Extended gcds and the Hermite normal form (Havas) Computing the Hermite normal form of integer matrices is impor- tant in many applications. We explain how HNF calculation can be viewed in terms of extended gcd computations. We present new, more effective algorithms for both extended gcd and HNF computa- tion. These algorithms can be formally analyzed. We present exam- ples showing their excellent performance. She'll be right, mate: French metaphysics (Playoust) The new PARI system modules in Magma include those for the real and complex fields, power series and Laurent series rings, and p-adic fields. This talk will outline the approach that has taken to subsuming PARI's element-orientation under Magma's structure- orientation. Of particular interest is the extent to which rela- tive precision is seen as an inherent part of the field model. Most of the examples will be concerned with the real and complex fields, and will illustrate Magma's new wealth of analytical functions. Surfing the lava flow: Constructing objects in Magma (Playoust) This talk will introduce the main techniques for constructing rings, fields and modules in Magma, together with elements of such structures. Recent additions to Magma will receive particu- lar attention, and the similarities across categories will be em- phasized. Following this talk, workshop participants should have sufficient confidence to experiment with Magma themselves. Fast factorization of univariate polynomials (Steel) We have recently installed in Magma a new improved algorithm for factoring univariate polynomials over the integers. It is based on the recent paper `Improved Techniques in Univariate Polynomial Factorization' by G. Collins and M. Encarnacion. Important use is made of early factor detection and the testing of combinations of modular factors rather than just single modular factors. Various extra heuristics are also employed. A general outline of the algorithm will be given together with its performance and ap- plication in other parts of Magma. Sharpening the meataxe: The structure of modules over Q (Steel) The `Meataxe' algorithm for splitting modules over finite fields is now being extended to work over fields with characteristic 0. An efficient algorithm for modules over Q has been developed which uses the finite field meataxe and converts a modular sub- space back to a subspace over Q. We discuss the algorithm and its application in other parts of Magma (e.g. finite dimensional associative algebras).