Exploring Abstract Algebra With Mathematica®

467 Pages·1999·0.78 MB·English

by  Allen C. Hibbard

EXPLORING
ABSTRACT ALGEBRA
WITH
MATHEMATICA®
Allen C. Hibbard
Kenneth M. Levasseur
EXPLORING
ABSTRACT ALGEBRA
WITH
MATHEMATICA
®
EXIRA
MATERIALS
extras.springer.com
Allen C. Hibbard
Department of Mathematics and Computer Science
Central College
Pella, IA 50219
USA
Kenneth M. Levasseur
Department of Mathematics and Computer Science
University of Massachusetts Lowell
Lowell, MA 01854
USA
Library of Congress Cataloging-in-Publication Data
Hibbard, Allen C.
Exploring abstract algebra with Mathematica I Allen C. Hibbard,
Kenneth M. Levasseur.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-387-98619-7 ISBN 978-1-4612-1530-1 (eBook)
DOI 10.1007/978-1-4612-1530-1
1. Mathematica (Computer file) 2. Algebra, Abstract-Data
processing. 1. Levasseur, Kenneth M. II. Title.
QA162.H52 1998
512'.02'028553042--dc21 98-41141
Printed on acid-free paper.
Additional material to this book can be downloaded from hHp://extras.springer.com
© 1999 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, loc. in 1999
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987654321
ISBN 978-0-387-98619-7
Preface
• What is Exploring Abstract Algebra with Mathematica?
Exploring Abstract Algebra with Mathematica is a learning environment for
introductory abstract algebra built around a suite of Mathematica packages enti
tled AbstractAlgebra. These packages are a foundation for this collection of
twenty-seven interactive labs on group and ring theory. The lab portion of this
book reflects the contents of the Mathematica-based electronic notebooks con
tained in the accompanying CD-ROM. Students can interact with both the printed
and electronic versions of the material in the laboratory and look up details and
reference information in the User's Guide. Exercises occur in the stream of the
text of labs, providing a context in which to answer. The notebooks are designed
so that the answers to the questions can either be entered into the electronic
notebook or written on paper, whichever the instructor prefers. The notebooks
support versions 2.2 and 3.0-4.0 and are compatible with all platforms that run
Mathematica.
This work can be used to supplement any introductory abstract algebra text and is
not dependent on any particular text. The group and ring labs have been cross
referenced against some of the more popular texts. This information can be found
on our web site at http://www . central. edu/eaarn. htrnl (which is also
mirrored at http://www . urnl. edu/Dept/Math/eaarn/eaarn. htrnl). If
your favorite text isn't on our list, it can be added upon request by contacting
either author. The AbstractAlgebra packages and the electronic documenta
tion files are freely available for downloading from our web page, as are a variety
of other resources (including useful palettes and ideas for further exploration). As
we add new functionality to the packages or supplement the documentation files,
vi Preface
updates will be made available at this web site and users are encouraged to sign
up there to be notified when updates are posted .
• More about the AbstractAlgebra packages
The AbstractAlgebra packages provide the Mathematica programming code
to work with structures in abstract algebra. Currently, the packages are capable of
handling most of the types of objects encountered in a first-year undergraduate
abstract algebra course. This includes working with (finite) groups, rings, fields,
and morphisms and functions related to each of these objects. There are a large
number of built-in groups (including such standard groups as Zn, Un (units of
Zn), Sn, Dn, as well as direct products and quotients of these) and rings (including
Zn' Boolean rings, and lattice rings, as well as polynomial, matrix, and function
extension rings). One can also create functions between groups or rings and
investigate if these are morphisms. Documentation for these packages forms the
second half of this book. After an introductory chapter, there is a chapter for
working with groups, one for rings and one for morphisms. A final chapter
focuses on other functions built into the packages. This portion of the book is
intended to be a reference for working with the AbstractAlgebra packages .
• More about the Mathematica labs
Exploring Abstract Algebra with Mathematica is intended for anyone trying to
learn (or teach) abstract algebra. This course is often challenging because of its
formal and abstract nature. While some people are quite adept at thinking
abstractly, many are helped by also thinking visually or geometrically. To this
end, where possible, the Mathematica labs are designed to appeal to the visualiza
tion of various algebraic ideas (as pioneered by Ladnor Geissinger in his software
package Exploring Small Groups). Additionally, the nature of the Mathematica
notebooks encourages an exploratory environment in which one can make and test
conjectures. Viewing the notebooks as interactive texts allows an environment
that cannot be replicated by lecture alone. While many of the labs are designed to
prepare the way for in-class discussion/lecture, they can also be used to extend
examples seen in class.
There is no assumption about being able to program in Mathematica; users only
need to know the basic concepts of using Mathematica, which are reviewed in
Lab 0 Getting Started with Mathematica (found in Appendix B). Every lab starts
with a set of goals as well as the prerequisites. Most labs are independent, though
Preface vii
a few assume some experience with a previous lab. Although Part I of this book
contains the 14 group labs followed by Part II containing the 13 ring labs, the ring
labs can just as easily be used first by those who prefer to do rings first (as one of
us does). Questions are interspersed throughout the lab at the points where it is
natural to ask them. As with any text, one does not need to complete every ques
tion in every lab. While the length of the labs varies from 40 minutes to 90 min
utes, they typically require about 60 minutes. (Of course, this is a function of how
many of the questions are answered.) For adopters of the book, there are provi
sions for suggested minimal questions to be assigned, as well as listing which
ones can be considered optional. Partial solutions are also available for instructors
upon request (bye-mailing either author).
The enclosed CD-ROM contains all the labs (as Mathematica notebooks), the
User's Guide, the AbstractAlgebra packages, and a number of palettes (for
users of version 3.0 or higher-to facilitate the use of the labs and the implementa
tion of the packages). While a copy of Mathematica is necessary to perform any
evaluations, several versions of the read-only MathReader are also included on
the CD-ROM. Instructions can be found both on the CD-ROM and in Appendix A
for installation of the packages, documentation files, and lab notebooks. Except
for the group and ring labs (and some palettes), the materials on the CD-ROM are
also on our web site. (The web site also has notebooks containing just the ques
tions that students can use as "answer sheets," as well as other resources.)
" Acknowledgments
We would like to thank the many people who have provided us with support for
this project. This includes Stan Seltzer and Connie Elson (both at Ithaca College),
whose NSF-sponsored workshop initially brought us together in 1992 to work on
this project. We appreciate being involved with the Interactive Mathematics Text
Project (IBMlMANNSF funded) which provided encouragement and support for
several summers in the early years of this project. We are thankful to Wolfram
Research, Inc., for providing technical tools, support staff, and the opportunity to
participate in their Visiting Scholars program. In particular, Brett Barnhart, Andre
Kuzniarek, and Paul Wellin were helpful on numerous occasions. We appreciate
the ongoing support of our institutions, Central College and University of Massa
chusetts Lowell. Furthermore, we appreciate our students for their patience while
testing these labs, and particularly Rochelle Rucker and Michael Thompson for
their steadfast assistance in testing, editing, proofing, and converting notebooks.
We are pleased with the folks at Springer-Verlag who helped bring this project to
a conclusion, particularly Keisha Sherbecoe for her attention to details and Steven
viii Preface
Pisano for his expertise and helpfulness. We also appreciate Stan Wagon (Macal
ester College) for initially encouraging the publication of this project with
Springer-Verlag. Our indebtedness also goes to the number of testers of the
packages and labs, particularly to Eric Gossett (Bethel College) and John Kiehl
(Soundtrack Recording Studios), as well as reviewers Tom Halverson (Macalester
College) and Garry Helzer (University of Maryland). Finally, we would like to
thank our families who have supported us throughout this project; Al particularly
appreciates his daughter Christina who provided constant companionship at his
desk while working, and Ken appreciates his wife Karen, and children, Joe,
Kathryn, and Matt for their encouragement.
Al Hibbard
[email protected]
Ken Levasseur
[email protected]
Contents
Preface .......................................................................... ....... v
PART I GroupLabs
1 Using Symmetry to Uncover a Group................................. 3
Getting started? Begin here • A symmetry of an equilateral triangle •
Are there other symmetries? • Multiplying the transformations • Are
there any commuters? • Is it always bad to be closed-minded? • We
should try to find our identity • Is it perverse to not have an inverse?
• Should we associate together? • What else? • Let's group it all
together
2 Determining the Symmetry Group of a Given Figure ........ 11
Symmetries and how to find them • Your turn
3 Is This a Group?................................................................. 17
When do we have a group? • Your turn
4 Let's Get These Orders Straight......................................... 20
Order of g and its inverse • Distribution of the orders of elements in
Zn • Another look at orders • What is P( I g 1= n) for g E Zn? •
More questions about Un
5 Subversively Grouping Our Elements................................ 32
When do we have a subgroup? • Subgroups of Zn • P(H < G) for a
=
random subset H of G Zn • Necessary elements for full closure •
Subgroups of Un
x Contents
6 Cycling Through the Groups.............................................. 45
What, when, how, and why about cyclic groups • Cyclicity of
Zm EB Zn • Structure of intersections of subgroups of Z
7 Permutations ....................................................................... 53
What is a permutation? • Computations with permutations •
Applications of permutations • Questions about permutations
8 I somorphisms....................................................................... 64
What is an isomorphism? • Creating Morphoids • Seeing
isomorphisms
9 A utomorphisms ................................... ....................... .... ..... 74
Automorphisms on Zn • Inner automorphisms
10 Direct Products.................................................................... 81
What is a direct product? • Order of an element in a direct product •
When is a direct product of cyclic groups cyclic? • Isomorphisms
among Un groups
11 Cosets................................................................................... 88
Cosets, left and right • Properties of cosets
12 Normality and Factor Groups............................................ 95
Normal subgroups • Making a new group • Factor groups
13 Group Homomorphisms ...... ............................................... 101
What is a group homomorphism? • The kernel and image •
Properties that are preserved by homomorphisms • The kernel is
normal • The First Homomorphism Theorem • The alternating
group-parity as a morphism
14 Rotational Groups of Regular Polyhedra.......................... 111
The rotational group of the tetrahedron • Further exercises
Contents xi
PART II Ring Labs
1 Introduction to Rings and Ringoids. .............................. 119
Getting started? Begin here • Ringoids and rings • Properties of
rings • Additional exercises
2.............................................. 127
2 Introduction to Rings, Part
Units and zero divisors • Integral domains • Fields • Additional
exercises
3 An Ideal Part of Rings. ....................................................... 134
What is the ideal part of a ring? • Ideals factor into other ring
properties
4 What Does Zbl/(a + b i) Look Like? ............................... 140
Example 1 • Example 2
5 Ring Homomorphisms ......................................... ....... ........ 148
Morphoids on rings • Ring homomorphisms • The kernel and
image • The kernel is an ideal • One-rule Morphoids • The
Chinese Remainder Theorem
6 Polynomial Rings ................................................................ 156
Introduction to polynomials • Divide and conquer
7 F aetoTing and I rreducibility ............................................... 167
Introduction to factoring and irreducibility • Some techniques on
testing the irreducibility of polynomials • More polynomials for
practice • Toolbox of theorems • Final perspective
Unity......................................................................
8 Roots of 182
Introduction • A closer look-graphically • Another
look-algebraically
9 Cyclotomic Polynomials ...................................................... 190
Introduction • Search for gn(x) • Some properties of <l>n(x)