Klima R., Sigmon N., Stitzinger E. - Applications of Abstract Algebra with MAPLE.pdf - Algebra - besia86

Applications of

Abstract Algebra

with MAPLE

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 .

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

1999 by CRC Press LLC

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