Klima R., Sigmon N., Stitzinger E. - Applications of Abstract Algebra with MAPLE.pdf
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Applications of
Abstract Algebra
with MAPLE
c
1999 by CRC Press LLC
Applications of
Abstract Algebra
with
MAPLE
Richard E. Klima
Neil Sigmon
Ernest Stitzinger
CRC Press
Boca Raton London New York Washington, D.C.
Preface
In 1990 we introduced a one-semester applications of algebra course at
North Carolina State University for students who had successfully com-
pleted semesters of linear and abstract algebra.We intended for the course
to give students more exposure to basic algebraic concepts, and to show
students some practical uses of these concepts.The course was received
enthusiastically by both students and faculty and has become one of the
most popular mathematics electives at NC State.
When we were originally deciding on material for the course, we knew
that we wanted to include several topics from coding theory, cryptography,
and counting (what we call
Polya
theory).With this in mind, at the sug-
gestion of Michael Singer, we used George Mackiw’s book
Applications of
Abstract Algebra
for the first few years, and supplemented as we saw fit.
After several years, Mackiw’s book went out of print temporarily.Rather
than search for a new book for the course, we decided to write our own notes
and teach the course from a coursepack.About the same time, NC State
incorporated the mathematics software package Maple V
TM
1
into its calcu-
lus sequence, and we decided to incorporate it into our course as well.The
use of Maple played a central role in the recent development of the course
because it provides a way for students to see realistic examples of the topics
discussed without having to struggle with extensive computations.With
additional notes regarding the use of Maple in the course, our coursepack
evolved into this book.In addition to the topics discussed in this book, we
have included a number of other topics in the course.However, the present
material has become the constant core for the course.
Our philosophy concerning the use of technology in the course is that
it be a useful tool and not present new problems or frustrations.Conse-
quently, we have included very detailed instructions regarding the use of
1
Maple V is a registered trademark of Waterloo Maple, Inc., 57 Erb St. W, Waterloo,
CanadaN2L6C2,
www.maplesoft.com
.
c
1999 by CRC Press LLC
Maple in this book.It is our hope that the Maple discussions are thorough
enough to allow it to be used without much alternative aid.As alterna-
tive aids, we have included a basic Maple tutorial in Appendix A, and an
introduction to some of Maple’s linear algebra commands in Appendix B.
Although we do not require students to produce the Maple code used in
the course, we do require that they obtain a level of proficiency such that
they can make basic changes to provided worksheets to complete numerous
Maple exercises.So that this book can be used for applications of algebra
courses in which Maple is not incorporated, we have separated all Maple
material into sections that are clearly labeled, and separated all Maple and
non-Maple exercises.
When teaching the course, we discuss the material in Chapter 1 as
needed rather than review it all at once.More specifically, we discuss the
material in Chapter 1 through examples the first time it is needed in the ap-
plications that follow.Some of the material in Chapter 1 is review material
that does not apply specifically to the applications that follow.However,
for students with weak backgrounds, Chapter 1 provides a comprehensive
review of all necessary prerequisite mathematics.
Chapter 2 is a short chapter on block designs.In Chapters 3, 4, and
5 we discuss some topics from coding theory.In Chapter 3 we introduce
error-correcting codes, and present Hadamard, Reed-Muller, and Hamming
codes.In Chapters 4 and 5, we present BCH codes and Reed-Solomon
codes.Each of these chapters are dependent in part on the preceding chap-
ters.The dependency of Chapter 3 on Chapter 2 can be avoided by omitting
Sections 3.2, 3.3, and 3.4 on Hadamard and Reed-Muller codes. In Chap-
ters 6, 7, and 8 we discuss some topics from cryptography.In Chapter 6
we introduce algebraic cryptography, and present several variations of the
Hill cryptosystem.In Chapter 7 we present the RSA cryptosystem and
discuss some related topics, including the Di<e-Hellman key exchange.In
Chapter 8 we present the ElGamal cryptosystem, and describe how elliptic
curves can be incorporated into the system naturally.There is a slight de-
pendency of Chapters 7 and 8 on Chapter 6, and of Chapter 8 on Chapter
7.Chapter 9 is a stand-alone chapter in which we discuss the Polya count-
ing techniques, including Burnside’s Theorem and the Polya Enumeration
Theorem.
We wish to thank all those who have been involved in the develop-
ment of this course and book.Pete Hardy taught from the coursepack and
improved it with his suggestions.Also, Michael Singer suggested various
topics and wrote notes on some of them.Many students have written on
this material for various projects.Of these, the recent master’s project by
Karen Klein on elliptic curves was especially interesting.Finally, we wish to
c
1999 by CRC Press LLC
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