Saint Louis University

Department of Mathematics and Mathematical Computer Science

Maple worksheets for abstract algebra

Overview

These worksheets are part of an ongoing project of mine to produce courseware for using Maple to help students learn concepts in abstract algebra. I want my students to have the power of Maple for exploring difficult concepts and for gaining intuition. On the other hand, I don't want my class to become a class in Maple. The model I am using is that Maple is a programming language that the instructor uses to produce specific user friendly applications that the student can use to explore a particular concept.

The worksheets are being developed with exercises to encourage the students not only to follow the Maple computations, but to experiment with variations from the worked examples. Nevertheless it is important that the worksheets be available to those who don't understand computers at all. Thus, the first time through the worksheet, the student only has to hit enter to see the results of the computations. The second time through, the student can use the worksheet as a template, changing numbers to explore similar problems. If the student finds the material interesting, he or she can use the code as a model for using Maple for more work. The image I have is not to create black boxes, but to create clear fiberglass boxes. The user can ignore the inner workings, but they are visible for inspection if the user wants to tinker.

This project started on Maple V release 3 on a Macintosh. They have been revised to run on Maple 9.5.  As I converted old worksheets and wrote new ones, I added exercises.  I find that this encourages the students to explore, and gain better insight. I have also added a number of references to the textbook (Abstract Algebra, by Dummit and Foote) we have been using.

• The first three worksheets are concerned with concepts and examples that come up in the study of rings.
  • The first worksheet, FactoringExamples.mw, is a preliminary worksheet to familiarize the student with the peculiarities of the factor command in Maple.
  • The second worksheet, GaussInt.mw, looks at standard ring concepts over the Gaussian integers. The worksheet looks at the standard Euclidean norm, prime factorization, the Euclidean algorithm, and the Chinese Remainder theorem. The worksheet utilizes the GaussInt package.
  • The third worksheet, QuadraticEuclidean.mw, looks at the this same norm over a number of other quadratic extensions of the integers. It considers extensions where the norm is Euclidean as well as extensions where the norm is not Euclidean. Then the norm is Euclidean, the students use it in the Euclidean algorithm.

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• The next two worksheets look at ideas that come up in the study of simple extensions of fields.
  • The worksheet MinPloy.mw explains a way to use Maple to find the minimum polynomial of an element that is algebraic over the rationals. The algorithm uses the Grobner package.
  • The worksheet ComputingInFields.mw shows how to compute the inverse of an element in a simple extension of the rational. (With inverses, one can also compute quotients.) The process uses the extended gcd command , and hence is an application of the Euclidean algorithm over the polynomial ring.

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• The next three worksheets look at automorphisms of field extensions, the heart of Galois theory.
  • The worksheet DefiningAutomorphisms.mw takes the student through the process of defining automorphisms of finite extensions of the rationals by defining actions on the generators of the extension. The question of when such a definition defines an automorphism is addressed.
  • The worksheet AutomorphismGroups.mw looks at finding the group structure of a set of automorphisms of a finite extension of the rationals. The worksheet explores finding the order of an automorphism and composing automorphisms. 
  • The worksheet AutosOverFiniteFields.mw looks at the group of automorphisms of a finite extension of a finite prime field. Since the fields involved are finite, these examples allow exhaustive testing. 

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• The next pair of worksheets look at the problem of finding the Galois group of an irreducible polynomial over the rationals.
  • The worksheet GaloisGroupOfPoly.mw breaks apart Maple's galois command and looks at the algorithm Maple uses to find the Galois group of an irreducible polynomial over the rationals of degree at most 7. The students then use the algorithm to find the Galois group of a polynomial of degree 8.
  • The worksheet GaloisPolys2.mw walks the students through a probabilistic argument that can be used to find the Galois group of a polynomial of arbitrary degree.