Renovating the Mathematics Curriculum: Exemplary Galois Theory

This proposal seeks ACS funding to complete the technology portion of an ongoing project: the creation of a course in undergraduate Galois theory which brings abstraction and computation together in the service of learning an important mathematical discipline. Such a course has a natural place in the mathematics curricula of many institutions, including 13 members of the ACS. Specifically, ACS funding during the summer of 2002 would support the design of packages in two computer algebra systems, Mathematica and Maple, to permit students to calculate complicated examples with ease. In addition, this project would involve ACS mathematics faculty in the review and evaluation of these materials. Taken in context of the larger project, the ACS grant would allow the proposer to finish the course development project, and the text would reach a publisher during the following year.

Yet practically since introductory texts in the subject became available in the 1930's and 1940's, the styles of presentation reflected contradictory fads in mathematics. A survey of these texts reveals, on one hand, Galois theory presented in as far-reaching a fashion as possible, taking the discipline to the limits of abstraction by presenting vast technical machinery that can be applied to a host of different problems. On the other hand, and in part in reaction to these former texts, we also find Galois theory presented in an unapologetically historical fashion, with a refusal to employ modern definitions and perspectives. This professional bickering, combined with the fact that until the 1980's, no significant tools were available to make computations in Galois theory reasonably accessible, simply left out of the loop those interested in Galois theory for undergraduate curricula: graduate schools of mathematics claimed Galois theory for their own and devised texts for it that suited their needs. Modern texts now assume a great deal of prior knowledge, and, even worse, assume a high level of "mathematical maturity," a degree of comfort with technical details and concepts which are only sketched instead of explained. The few exceptions which follow a historical tradition are necessarily limited by the miniscule number of computations one can actually perform by hand, and these exceptions prove the rule. The most recent text, Jean-Pierre Escofier's Théorie de Galois: Cours avec exercices corrigés, translated into English last year, confirms the disappointment of instructors at undergraduate institutions: designed for second-cycle and third-cycle French students, it assumes a preparation far beyond that of an American undergraduate.

Restoring Galois theory to its deserved place in the undergraduate mathematics curriculum, less as a monument to its former glory than as an exciting point of departure for the many directions taken by contemporary Galois theorists, requires fashioning a new course, designed from the bottom up with undergraduates in mind. The course must present the discipline stripped of the baggage of generality and reduced to its original context---the study of roots of polynomials with rational coefficients. From there, however, in place of a historical approach to the concepts and results, the course must interweave a consideration of modern, computational techniques into the presentation. A computational perspective is necessary for several reasons: to attract students into the course; to encourage intuition concerning intermediate results in the course; and, equally as importantly, to put into the hands of the students the ability to create and manipulate examples. This Galois theory course, more than any of its predecessors, must not only present examples but also must present many of the means by which those examples are constructed. Undergraduates will then come to grips with the abstract concepts in a pedagogically sound framework, asking questions and answering them by means of a large collection of examples.

Description: Proposed use of an ACS Teaching with Technology fellowship. The proposer seeks a summer ACS Teaching with Technology Fellowship in order to improve and complete the computational underpinnings for this course. ACS funding would permit the allocation of summer research time to the improvement and completion of software materials for both an ACS audience and then a national audience.

The proposer will improve and complete a set of functions to be made available to students in the form of a package for a preexisting computer algebra system. Nearly every department of mathematics has available to its students one of two computer algebra systems which together nearly dominate the market for symbolic algebra tools, Mathematica (Wolfram Research) and Maple (Waterloo Maple Software), and hence packages will be developed for each system. The proposer will survey recent algorithms for Galois-theoretic computations and develop them for pedagogical use, perhaps leaving out the fastest algorithms if their intermediate results and approach yield little information for undergraduates--encouraging the idea that computation systems are "black boxes." The proposer will implement those algorithms which can also be presented to students in the text, achieving a balance between speed and classroom value.

Moreover, the proposer will take the time necessary to create an intuitive and user-friendly interface for these functions; the means by which data is entered, computations requested, and results tabulated will present as little an obstruction as possible. At present, the proposer's drafts of the packages now provide a minimal set of functions in an interface which is less than optimal. As a practical matter of teaching during the academic year, it has been more important with each instance of the course to implement new algorithms rather than find the time to revamp the interface from the ground up.

Finally, the proposer will explore the possibility that an even cleaner interface may be possible using webMathematica, a package from Wolfram Research released only in September 2001, which provides tools for implementing computer algebra system functions using a web interface. As a result, students would not need to run computer algebra systems at their home institutions but could use their web browsers to perform fast and instructive calculations. If the functions could be successfully implemented using this software, and if one or more institutions might agree to run the software on their web servers, then students at nearly every institution would be able to use the technology without the institutions' or the students' needing to purchase computer algebra systems. The proposer will acquire webMathematica and evaluate the suitability of transferring this project to that platform.

Timeline: Deliverables/milestones for the ACS-funded portion

Spring 2002: webMathematica acquired for testing and evaluation

June 2002: Work on design and interface begins in earnest

August 2002: Work on design and interface completed; if suitable, transfer entire interface to webMathematica server; otherwise, complete development of intuitive interface for packages in Mathematica and Maple

September 2002: Text and packages disseminated to evaluators at ACS institutions and elsewhere

Spring 2003: After modifications based on comments of evaluators, packages are made freely available over the web to ACS institutions and elsewhere; text reaches publisher

Technical requirements for the project. The project requires current versions of all software mentioned thus far, and, if the webMathematica platform proves optimal, a server as well. The proposer has all software and would be able to procure webMathematica using institutional funds. Use of a server, if necessary, may be obtained by employing a departmental computer as a server temporarily or by requesting temporary use of another department's server.

In addition, the proposer needs technical skill in programming in each software package. The proposer has already gained such skills for Mathematica and Maple (having, in particular, discovered errors in previous releases of Mathematica) and will gain them quickly for webMathematica.

Other support. The first stage of the textbook project was partially supported by the National Science Foundation. Support for this last stage of the project is a portion of a larger grant proposal currently under consideration by the National Science Foundation.

Learning outcomes. This project will aid both instructors and students in probing an important mathematical discipline of current interest by providing a great number of examples from which conjectures may be derived and intuition deepened. Existing texts for this discipline provide little or no computational support and, as a result, discourage the development of intuition through example. Nearly every aspect of this course, however, will be supported by computations with the packages under development. These packages will enable the solution of a variety of exercises aimed at generating further questions and solidifying intuition into comprehension. This project does more than improve a portion of an existing course; it places an entire course on an entirely new footing, one which has undergraduates as the primary audience.

Curriculum. A second course in abstract algebra is taught in many departments of mathematics, sometimes on an alternate-year schedule. As a result, the topic of this course is already integrated in to curricula in many mathematics departments. The new course could be adopted without difficulty in curricula which already offer a second course in algebra, and most departments of mathematics would especially welcome such a course which integrates computational tools with the study of an advanced topic. Considering ACS departments specifically, fully thirteen offer courses which could incorporate an undergraduate text in Galois theory, at least as a module for a general second course in algebra (Birmingham-Southern: MA 452; Centre: MAT 331; Davidson: MAT 455; Furman: MTH 44; Hendrix: MATH 420; Rhodes: Math 363; Richmond: Math 307; Rollins: MAT 480; Sewanee: Math 306; Southwestern: 52-693; Spelman: MATH 472; Trinity: MATH 4363; Washington and Lee: MATH 322), and indeed some catalog descriptions devote the second course in algebra entirely to Galois theory.

Assessment. The project as a whole, and the technology portion in particular, are being and will continue to be assessed by colleagues in mathematics departments at other institutions, particularly ACS institutions. Individuals who have expressed interest in the project, whether to consider using the materials or to evaluate them or both, are as follows (asterisks denote ACS faculty): Elizabeth Allman (University of Southern Maine); David Bressoud (Macalester College); Ezra Brown (Virginia Tech); Blayne Carroll (Berry College); Wayne Dymacek* (Washington and Lee University); Griff Elder (University of Nebraska at Omaha); Alex McAllister* (Centre College); Walt Potter* (Southwestern University); Doug Rall* (Furman University); David Robbins (Trinity College, CT); John Wilson* (Centre College).

Dissemination. In accordance with the timeline above, Mathematica and Maple packages will be made freely available over the web to all institutions. If used, webMathematica-based material will be available to all institutions, requiring only a web browser. In addition, the proposer will make technical support particularly available to colleagues at ACS institutions. The proposer will provide drafts of texts to evaluators and other ACS colleagues as the project progresses. Eventually texts will be disseminated through a publisher.