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Full Description
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.
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
