The "Magma" software was acquired for use in a class on algebraic number theory (fall, '01) and in subsequent student thesis work. In fact, the package has been a big success even outside those boundaries: I have used it in subsequent courses and in my research; in addition, a thesis student this year is using it.
Magma is designed to provide efficient algorithms for the advanced aspects of number fields: number theory, algebra, combinatorics, and geometry. It is a successor to a package called "Cayley'' that was developed from 1975 through 1990 for group theory and representation theory; the experience in the advantages of a "structural'' approach to symbolic mathematics led the computational algebra team to begin a vast upgrade in 1993 to include the areas of mathematics named above.
The Magma team has been very successful in attracting number theorists, algebraic geometers, and others to their group, and the Magma package is far and away the best available package for many purposes. Almost all of the sophisticated computational researchers in the areas in which I now work use this tool. As a consequence of its growing prominence, there are several books in preparation describing the current program.
Algebraic number theory is usually taught at the graduate level, and is a fascinating juxtaposition of abstract ideas arising from concrete questions in number theory. Anyone that learns the subject needs to develop a feeling for the abstractions by learning to do computations.
It is unusual to use computation in a systematic fashion in a theoretical mathematics course at this level. Because of the value of concrete examples, it seemed appropriate to incorporate Magma in the course in fall '01. In fact, this worked smoothly, and my sense was that the class appreciated Magma's ability to construct and discover examples far beyond what could be done by hand. It seemed to be especially useful for students to be able to study very abstruse concepts in class, and then be amazed by the power and generality of the tools available in Magma for manipulating those abstractions.
One indication of the utility of Magma has been the fact that students "voted with their fingers;'' several students in the class have been using it ever since. Two of the students who were seniors used Magma in their thesis research one way or another in spring '02, and several other faculty have used the program in their mathematics. In addition, I have been delighted to use Magma in several areas of my research, and have found it to be absolutely crucial in two key instances. Finally, a current thesis student (who took algebraic number theory last year as a junior) is working on a thesis that grows out of algebraic number theory; both of us have found Magma to be very useful at several points in the project, and I anticipate that this will continue to be true this spring.