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.

Table of Contents:
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