The Third Australian Magma Workshop (Macquarie University, 18 Apr, 1995)

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:

  1. To inform people of the ring, field and module theory capabilities;
  2. To provide a basic tutorial in using the system;
  3. To outline some of the ideas behind new algorithms that have very recently been incorporated into Magma.

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.


 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



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-

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

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).