Introduction to Abstract Algebra, 4e Set

This Fourth Edition of Introduction to Abstract Algebra is a self-contained introduction to the basic structures of abstract algebra: groups, rings, and fields. This book is intended for a one or two semester abstract algebra course. The writing style is appealing to students, and great effort has been made to motivate and be very clear about how the topics and applications relate to one another. Over 500 solved examples are included to aid reader comprehension as well as to demonstrate how results in the theory are obtained. Many applications (particularly to coding theory, cryptography, and to combinatorics) are provided to illustrate how the abstract structures relate to real-world problems. In addition, historical notes and biographies of mathematicians put the subject into perspective. Abstract thinking is difficult when first encountered and this is addressed in this book by presenting concrete examples (induction, number theory, integers modulo n, permutations) before the abstract structures are defined. With this approach, readers can complete computations immediately using concepts that will be seen again later in the abstract setting. Special topics such as symmetric polynomials, nilpotent groups, and finite dimensional algebras are also discussed.

W. KEITH NICHOLSON, PhD, is Professor in the Department of Mathematics and Statistics at the University of Calgary, Canada. He has published extensively in his areas of research interest, which include clean rings, morphic rings and modules, and quasi-morphic rings. Dr. Nicholson is the coauthor of Modern Algebra with Applications, Second Edition, also published by Wiley.

Preface ix

Acknowledgment xv

Notations Used in the Text xvii

A Sketch of the History of Algebra to 1929 xxi

Preliminaries 1

Proofs 1

Sets 5

Mappings 9

Equivalences 17

Integers and Permutations 22

Induction 22

Divisors and Prime Factorization 30

Integers Modulo n 41

Permutations 51

An Application to Cryptography 63

Groups 66

Binary Operations 66

Groups 73

Subgroups 82

Cyclic Groups and the Order of an Element 87

Homomorphisms and Isomorphisms 95

Cosets and Lagrange's Theorem 105

Groups of Motions and Symmetries 114

Normal Subgroups 119

Factor Groups 127

The Isomorphism Theorem 133

An Application to Binary Linear Codes 140

Rings 155

Examples and Basic Properties 155

Integral Domains and Fields 166

Ideals and Factor Rings 174

Homomorphisms 183

Ordered Integral Domains 193

Polynomials 196

Polynomials 196

Factorization of Polynomials over a Field 209

Factor Rings of Polynomials over a Field 222

Partial Fractions 231

Symmetric Polynomials 233

Formal Construction of Polynomials 243

Factorization in Integral Domains 246

Irreducibles and Unique Factorization 247

Principal Ideal Domains 259

Fields 268

Vector Spaces 269

Algebraic Extensions 277

Splitting Fields 285

Finite Fields 293

Geometric Constructions 299

The Fundamental Theorem of Algebra 304

An Application to Cyclic and BCH Codes 305

Modules over Principal Ideal Domains 318

Modules 318

Modules over a PID 327

p-Groups and the Sylow Theorems 341

Factors and Products 341

Cauchy's Theorem 349

Group Actions 356

The Sylow Theorems 364

Semidirect Products  371

An Application to Combinatorics 375

Series of Subgroups 381

The Jordan-Holder Theorem 382

Solvable Groups 387

Nilpotent Groups 394

Galois Theory 401

Galois Groups and Separability 402

The Main Theorem of Galois Theory 410

Insolvability of Polynomials 423

Cyclotomic Polynomials and Wedderburn's Theorem 430

Finiteness Conditions for Rings and Modules 435

Wedderburn's Theorem 435

The Wedderburn-Artin Theorem 444

Appendices

Complex Numbers 455

Matrix Arithmetic 462

Zorn's Lemma 467

Proof of the Recursion Theorem 471

Bibliography 473

Selected Answers 475

Index 499