Abstract Algebra: A Gentle Introduction

214 Pages·2016·4.543 MB·English

by  Gary L. Mullen, James A. Sellers

ABSTRACT
ALGEBRA
A GENTLE INTRODUCTION
TEXTBOOKS in MATHEMATICS
Series Editors: Al Boggess and Ken Rosen
PUBLISHED TITLES
ABSTRACT ALGEBRA: AN INTERACTIVE APPROACH, SECOND EDITION
William Paulsen
ABSTRACT ALGEBRA: AN INQUIRY-BASED APPROACH
Jonathan K. Hodge, Steven Schlicker, and Ted Sundstrom
ADVANCED LINEAR ALGEBRA
Hugo Woerdeman
APPLIED ABSTRACT ALGEBRA WITH MAPLE™ AND MATLAB®, THIRD EDITION
Richard Klima, Neil Sigmon, and Ernest Stitzinger
APPLIED DIFFERENTIAL EQUATIONS: THE PRIMARY COURSE
Vladimir Dobrushkin
COMPUTATIONAL MATHEMATICS: MODELS, METHODS, AND ANALYSIS WITH MATLAB® AND MPI,
SECOND EDITION
Robert E. White
DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE, SECOND EDITION
Steven G. Krantz
DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE WITH BOUNDARY VALUE PROBLEMS
Steven G. Krantz
DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES, THIRD EDITION
George F. Simmons
DIFFERENTIAL EQUATIONS WITH MATLAB®: EXPLORATION, APPLICATIONS, AND THEORY
Mark A. McKibben and Micah D. Webster
ELEMENTARY NUMBER THEORY
James S. Kraft and Lawrence C. Washington
EXPLORING CALCULUS: LABS AND PROJECTS WITH MATHEMATICA®
Crista Arangala and Karen A. Yokley
EXPLORING LINEAR ALGEBRA: LABS AND PROJECTS WITH MATHEMATICA®
Crista Arangala
PUBLISHED TITLES CONTINUED
GRAPHS & DIGRAPHS, SIXTH EDITION
Gary Chartrand, Linda Lesniak, and Ping Zhang
INTRODUCTION TO ABSTRACT ALGEBRA, SECOND EDITION
Jonathan D. H. Smith
INTRODUCTION TO MATHEMATICAL PROOFS: A TRANSITION TO ADVANCED MATHEMATICS, SECOND EDITION
Charles E. Roberts, Jr.
INTRODUCTION TO NUMBER THEORY, SECOND EDITION
Marty Erickson, Anthony Vazzana, and David Garth
LINEAR ALGEBRA, GEOMETRY AND TRANSFORMATION
Bruce Solomon
MATHEMATICAL MODELLING WITH CASE STUDIES: USING MAPLE™ AND MATLAB®, THIRD EDITION
B. Barnes and G. R. Fulford
MATHEMATICS IN GAMES, SPORTS, AND GAMBLING–THE GAMES PEOPLE PLAY, SECOND EDITION
Ronald J. Gould
THE MATHEMATICS OF GAMES: AN INTRODUCTION TO PROBABILITY
David G. Taylor
A MATLAB® COMPANION TO COMPLEX VARIABLES
A. David Wunsch
MEASURE THEORY AND FINE PROPERTIES OF FUNCTIONS, REVISED EDITION
Lawrence C. Evans and Ronald F. Gariepy
NUMERICAL ANALYSIS FOR ENGINEERS: METHODS AND APPLICATIONS, SECOND EDITION
Bilal Ayyub and Richard H. McCuen
ORDINARY DIFFERENTIAL EQUATIONS: AN INTRODUCTION TO THE FUNDAMENTALS
Kenneth B. Howell
RISK ANALYSIS IN ENGINEERING AND ECONOMICS, SECOND EDITION
Bilal M. Ayyub
SPORTS MATH: AN INTRODUCTORY COURSE IN THE MATHEMATICS OF SPORTS SCIENCE AND
SPORTS ANALYTICS
Roland B. Minton
TRANSFORMATIONAL PLANE GEOMETRY
Ronald N. Umble and Zhigang Han

TEXTBOOKS in MATHEMATICS
ABSTRACT
ALGEBRA
A GENTLE INTRODUCTION
Gary L. Mullen
The Pennsylvania State University

University Park, USA
James A. Sellers
The Pennsylvania State University

University Park, USA
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2017 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed on acid-free paper
Version Date: 20161121
International Standard Book Number-13: 978-1-4822-5006-0 (Hardback)
This book contains information obtained from authentic and highly regarded sources. Reasonable
efforts have been made to publish reliable data and information, but the author and publisher cannot
assume responsibility for the validity of all materials or the consequences of their use. The authors and
publishers have attempted to trace the copyright holders of all material reproduced in this publication
and apologize to copyright holders if permission to publish in this form has not been obtained. If any
copyright material has not been acknowledged please write and let us know so we may rectify in any
future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,
transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or
hereafter invented, including photocopying, microfilming, and recording, or in any information
storage or retrieval system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access
www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc.
(CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization
that provides licenses and registration for a variety of users. For organizations that have been granted
a photocopy license by the CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and
are used only for identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
Contents

Preface ix

1 Elementary Number Theory 1

1.1 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Primes and factorization . . . . . . . . . . . . . . . . . . . . 6

1.3 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Solving congruences . . . . . . . . . . . . . . . . . . . . . . . 15

1.5 Theorems of Fermat and Euler . . . . . . . . . . . . . . . . . 20

1.6 RSA cryptosystem . . . . . . . . . . . . . . . . . . . . . . . . 28

2 Groups 33

2.1 Definition of a group . . . . . . . . . . . . . . . . . . . . . . 33

2.2 Examples of groups . . . . . . . . . . . . . . . . . . . . . . . 34

2.3 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.4 Cosets and Lagrange’s Theorem . . . . . . . . . . . . . . . . 49

3 Rings 55

3.1 Definition of a ring . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 Subrings and ideals . . . . . . . . . . . . . . . . . . . . . . . 61

3.3 Ring homomorphisms . . . . . . . . . . . . . . . . . . . . . . 63

3.4 Integral domains . . . . . . . . . . . . . . . . . . . . . . . . . 65

4 Fields 67

4.1 Definition and basic properties of a field . . . . . . . . . . . . 67

5 Finite Fields 73

5.1 Number of elements in a finite field . . . . . . . . . . . . . . 73

5.2 How to construct finite fields . . . . . . . . . . . . . . . . . . 75

5.3 Properties of finite fields . . . . . . . . . . . . . . . . . . . . 82

5.4 Polynomials over finite fields . . . . . . . . . . . . . . . . . . 86

5.5 Permutation polynomials . . . . . . . . . . . . . . . . . . . . 89

5.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.6.1 Orthogonal Latin squares . . . . . . . . . . . . . . . . 91

5.6.2 Diffie/Hellman key exchange . . . . . . . . . . . . . . 94

vii
viii Contents
6 Vector Spaces 99

6.1 Definition and examples . . . . . . . . . . . . . . . . . . . . . 99

6.2 Basic properties of vector spaces . . . . . . . . . . . . . . . . 103

6.3 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7 Polynomials 111

7.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.2 Unique factorization . . . . . . . . . . . . . . . . . . . . . . . 115

7.3 Polynomials over the real and complex numbers . . . . . . . 117

7.4 Root formulas . . . . . . . . . . . . . . . . . . . . . . . . . . 118

8 Linear Codes 127

8.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

8.2 Hamming codes . . . . . . . . . . . . . . . . . . . . . . . . . 132

8.3 Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

8.4 Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

8.5 Further study . . . . . . . . . . . . . . . . . . . . . . . . . . 143

8.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

9 Appendix 149

9.1 Mathematical induction . . . . . . . . . . . . . . . . . . . . . 149

9.2 Well-ordering Principle . . . . . . . . . . . . . . . . . . . . . 152

9.3 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

9.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

9.5 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

9.6 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

9.7 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . 165

10 Hints and Partial Solutions to Selected Exercises 167

Bibliography 195

Index 199
Preface

Theaimof thistextbook istoprovideabrief introductiontoabstract algebra.
We use the term “A Gentle Introduction” because we do not go into great
depth while covering various topics in abstract algebra. Instead, we provide
the reader with coverage of numerous algebraic topics to give the reader a
flavor of what we believe to be the most important areas of abstract algebra.
For example,we do notcovertopics such asthe Sylow Theorems in group
theory. Instead, we lead the reader through a proof of Lagrange’s Theorem,
and then in Chapter 8, we provide a beautiful application of this famous
group theory result to the study of error-correcting codes used so frequently
in today’s modern communications.
We also discuss the famous RSA cryptographic system and the
Diffie/Hellman key exchange for the secure transmission of information. In
addition, we discuss some fascinating ideas in the study of sets of mutually
orthogonal latin squares, an area of study that dates back to Leonard Euler
in 1782.
The text is intended to be used for a one-semester course in abstract
algebra. It is suitable for an introductory course in abstract algebra for math­
ematics majors. The text is also very suitable for education majors who need
to have an introduction to various topics in abstract algebra.
Theorems, definitions, examples, corollaries, etc., are all numbered con­
secutively within each chapter. For example, in the first chapter, item 1.3 is
Theorem 1.3, item 1.5 is Example 1.5, and item 1.8 is Corollary 1.8.
InChapter9,entitled “Appendix,”weprovided abrief reviewof numerous
topics with which the student may not be familiar. This chapter is also meant
for the students who need to refresh their memory on a few topics. The topics
covered in the Appendix include mathematical induction, the well-ordering
principle, sets, functions, permutations, matrices, and complex numbers.
Numerous exercises are provided at the end of almost every section. In
addition, in Chapter 10, entitled “Hints and Partial Solutions to Selected
Exercises,”we provide solutions to the odd-numbered problems, leaving the
even-numbered problems for homework use by the instructor. Exercises are
numbered using the system Exercise i.j.k. This refers to the k-th exercise in
Chapter i, Section j.
Wewould liketothankSergeBallifforhishelpinsettinguptheLaTeXfiles
forourmanuscript.GaryMullenwouldalso liketothankhiswife(BevMullen)
forherpatienceand understandingduring thewriting of thistext.Thanksare
ix