
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.
 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.
 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.
 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.
You can
also download a zipped archive
of the Abstract Algebra worksheets.
Comments
and feedback are appreciated. If you find the worksheets
useful, please email me at maymk@slu.edu.
Return to Fr. May's home page
Return to main
Courseware page.
Last updated 12/1/05
