Introduction to Abstract Algebra E-Book

W. Keith Nicholson

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 1598

Veröffentlichungsjahr: 2012

Beschreibung

Praise for the Third Edition

". . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . ."—Zentralblatt MATH

The Fourth Edition of Introduction to Abstract Algebra continues to provide an accessible approach to the basic structures of abstract algebra: groups, rings, and fields. The book's unique presentation helps readers advance to abstract theory by presenting concrete examples of induction, number theory, integers modulo n, and permutations before the abstract structures are defined. Readers can immediately begin to perform computations using abstract concepts that are developed in greater detail later in the text.

The Fourth Edition features important concepts as well as specialized topics, including:

• The treatment of nilpotent groups, including the Frattini and Fitting subgroups

• Symmetric polynomials

• The proof of the fundamental theorem of algebra using symmetric polynomials

• The proof of Wedderburn's theorem on finite division rings

• The proof of the Wedderburn-Artin theorem

Throughout the book, worked examples and real-world problems illustrate concepts and their applications, facilitating a complete understanding for readers regardless of their background in mathematics. A wealth of computational and theoretical exercises, ranging from basic to complex, allows readers to test their comprehension of the material. In addition, detailed historical notes and biographies of mathematicians provide context for and illuminate the discussion of key topics. A solutions manual is also available for readers who would like access to partial solutions to the book's exercises.

Introduction to Abstract Algebra, Fourth Edition is an excellent book for courses on the topic at the upper-undergraduate and beginning-graduate levels. The book also serves as a valuable reference and self-study tool for practitioners in the fields of engineering, computer science, and applied mathematics.

Leseprobe

Contents

Cover

Title Page

Copyright

Preface

Acknowledgments

Notation Used in the Text

A Sketch of the History of Algebra to 1929

Chapter 0: Preliminaries

0.1 Proofs

0.2 Sets

0.3 Mappings

0.4 Equivalences

Chapter 1: Integers and Permutations

1.1 Induction

1.2 Divisors and Prime Factorization

1.3 Integers Modulo n

1.4 Permutations

1.5 An Application to Cryptography

Chapter 2: Groups

2.1 Binary Operations

2.2 Groups

2.3 Subgroups

2.4 Cyclic Groups and the Order of an Element

2.5 Homomorphisms and Isomorphisms

2.6 Cosets and Lagrange's Theorem

2.7 Groups of Motions and Symmetries

2.8 Normal Subgroups

2.9 Factor Groups

2.10 The Isomorphism Theorem

2.11 An Application to Binary Linear Codes

Chapter 3: Rings

3.1 Examples and Basic Properties

3.2 Integral Domains and Fields

3.2 Exercises

3.3 Ideals and Factor Rings

3.4 Homomorphisms

3.5 Ordered Integral Domains

Chapter 4: Polynomials

4.1 Polynomials

4.2 Factorization of Polynomials over a Field

4.3 Factor Rings of Polynomials over a Field

4.4 Partial Fractions

4.5 Symmetric Polynomials

4.6 Formal Construction of Polynomials

Chapter 5: Factorization in Integral Domains

5.1 Irreducibles and Unique Factorization

5.2 Principal Ideal Domains

Chapter 6: Fields

6.1 Vector Spaces

6.2 Algebraic Extensions

6.3 Splitting Fields

6.4 Finite Fields

6.5 Geometric Constructions

6.6 The Fundamental Theorem of Algebra

6.7 An Application to Cyclic and BCH Codes

Chapter 7: Modules over Principal Ideal Domains

7.1 Modules

7.2 Modules Over a PID

Chapter 8: p-Groups and the Sylow Theorems

8.1 Products and Factors

8.2 Cauchy's Theorem

8.3 Group Actions

8.4 The Sylow Theorems

8.5 Semidirect Products

8.6 An Application to Combinatorics

Chapter 9: Series of Subgroups

9.1 The Jordan–Hölder Theorem

9.2 Solvable Groups

9.3 Nilpotent Groups

Chapter 10: Galois Theory

10.1 Galois Groups and Separability

10.2 The Main Theorem of Galois Theory

10.3 Insolvability of Polynomials

10.4 Cyclotomic Polynomials and Wedderburn's Theorem

Chapter 11: Finiteness Conditions for Rings and Modules

11.1 Wedderburn's Theorem

11.2 The Wedderburn–Artin Theorem

Appendices

Appendix A Complex Numbers

Appendix B Matrix Algebra

Appendix C Zorn's Lemma

Appendix D Proof of the Recursion Theorem

Bibliography

Selected Answers

Index

End User License Agreement

Copyright 2012 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Nicholson, W. Keith.

Introduction to abstract algebra / W. Keith Nicholson. –4th ed.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-118-13535-8 (cloth)

1. Algebra, Abstract. I. Title.

QA162.N53 2012

512′.02–dc23

2011031416

Acknowledgments

I express my appreciation to the following people for their useful comments and suggestions for the first edition of the book: F. Doyle Alexander, Stephen F. Austin State University; Steve Benson, Saint Olaf College; Paul M. Cook II, Furman University; Ronald H. Dalla, Eastern Washington University; Robert Fakler, University of Michigan–Dearborn; Robert M. Guralnick, University of Southern California; Edward K. Hinson, University of New Hampshire; Ron Hirschorn, Queen's University; David L. Johnson, Lehigh University; William R. Nico, California State University–Hayward; Kimmo I. Rosenthal, Union College; Erik Shreiner (deceased), Western Michigan University; S. Thomeier, Memorial University; and Marie A. Vitulli, University of Oregon.

I also want to thank all the readers who informed me about typographical and other minor errors in the third edition. Particular thanks go to:

Carl Faith, Rutgers University, for giving the book a careful study and making many very useful suggestions, too numerous to list here;

David French, Derbyshire, UK, for pointing out several typographical errors;

Michel Racine, Université d'Ottawa, for pointing out a mistake in an exercise deducing the commutativity of addition in a ring from the other axioms;

Yoji Yoshii, Université d'Ottawa, for revealing two errors in the exercises for Chapter 5;

Yiqiang Zhou, Memorial University of Newfoundland, for many helpful suggestions and comments.

For the fourth edition, special thanks go to:

Jerome Lefebvre, University of Ottawa, for pointing out several typographical errors;

Edgar Goodaire and his students, Memorial University, for finding dozens of typographical errors and making many useful suggestions;

Keith Conrad, University of Connecticut, for many useful comments on the exposition;

Nazih Nahlus, American University of Beirut, for the proof that a finite multiplicative group of a field is cyclic;

Matthew Greenberg, University of Calgary, for pointing out that Burnside's lemma on Counting Orbits was due to Cauchy and Frobenius.

Milosz Kosmider, student, for correcting an error in Chapter 0;

Yannis Avrithis, National Technical University of Athens, for pointing out dozens of typographical errors and making several suggestions.

It is a pleasure to thank Steve Quigley for his generous assistance throughout the project. Thanks also go to the production staff at Wiley and particularly to Susanne Steitz-Filler for keeping the project on schedule and responding so quickly to all my questions. I also want to thank Joanne Canape for her vital assistance with the computer aspects of the project.

Finally, I want to thank my wife, Kathleen, for her unfailing support. Without her understanding and cooperation during the many hours that I was absorbed with this project, this book would not exist.

Notation used in the Text

A Sketch of the History of Algebra to 1929

2500 BC Hieroglyphic numerals used in Egypt.

2400 BC Babylonians begin positional algebraic notation.

600 BC Pythagoreans discuss prime numbers.

250 Diophantus writes Arithmetica, using notation from which modern notation evolved, and insists on exact solutions of equations in integers.

830 al-Khowarizmi writes Al-jabr, a textbook giving rules for solving linear and quadratic equations.

1202 Leonardo of Pisa writes Liber abaci on arithmetic and algebraic equations.

1545 Tartaglia solves the cubic, and Cardano publishes the result in his Ars Magna. Imaginary numbers are suggested.

1580 Viète uses vowels to represent unknown quantities, with consonants for constants.

1629 Fermat becomes the founder of the modern theory of numbers.

1636 Fermat and Descartes invent analytic geometry, using algebra in geometry.

1749 Euler formulates the fundamental theorem of algebra.

1771 Lagrange solves the general cubic and quartic by considering permutations of the roots.

1799 Gauss publishes his first proof of the fundamental theorem of algebra.

1801 Gauss publishes his Disquisitiones Arithmeticae.

1813 Ruffini claims that the general quintic cannot be solved by radicals.

1824 Abel proves that the general quintic cannot be solved by radicals.

1829 Galois introduces groups of substitutions.

1831 Galois sends his great memoir to the French Académie, but it is rejected.

1843 Hamilton discovers the quaternions.

1846 Kummer invents his ideal numbers.

1854 Cayley introduces the multiplication table of a group.

1870 Jordan publishes his monumental Traité, which explains Galois theory, develops group theory, and introduces composition series.

1870 Kronecker proves the fundamental theorem of finite abelian groups.

1872 Sylow presents his results on what are now called the Sylow theorems.

1878 Cayley proves that every finite group can be represented as a group of permutations.

1879 Dedekind defines algebraic number fields, studies the factorization of algebraic integers into primes, and introduces the concept of an ideal.

1889 Peano formulates his axioms for the natural numbers.

1889 Hölder completes the proof of the Jordan–Hölder theorem.

1905 Wedderburn proves that finite division rings are commutative.

1908 Wedderburn proves his structure theorem for finite dimensional algebras with no nilpotent ideals.

1921 Noether publishes her influential paper on chain conditions in ring theory.

1927 Artin extends Wedderburn's 1908 paper to rings with the descending chain condition.

1929 Noether establishes the modern approach to the theory of representations of finite groups.