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.
表中的内容
PREFACE ix
ACKNOWLEDGMENTS xvii
NOTATION USED IN THE TEXT xix
A SKETCH OF THE HISTORY OF ALGEBRA TO 1929 xxiii
0 Preliminaries 1
0.1 Proofs / 1
0.2 Sets / 5
0.3 Mappings / 9
0.4 Equivalences / 17
1 Integers and Permutations 23
1.1 Induction / 24
1.2 Divisors and Prime Factorization / 32
1.3 Integers Modulo n / 42
1.4 Permutations / 53
1.5 An Application to Cryptography / 67
2 Groups 69
2.1 Binary Operations / 70
2.2 Groups / 76
2.3 Subgroups / 86
2.4 Cyclic Groups and the Order of an Element / 90
2.5 Homomorphisms and Isomorphisms / 99
2.6 Cosets and Lagrange’s Theorem / 108
2.7 Groups of Motions and Symmetries / 117
2.8 Normal Subgroups / 122
2.9 Factor Groups / 131
2.10 The Isomorphism Theorem / 137
2.11 An Application to Binary Linear Codes / 143
3 Rings 159
3.1 Examples and Basic Properties / 160
3.2 Integral Domains and Fields / 171
3.3 Ideals and Factor Rings / 180
3.4 Homomorphisms / 189
3.5 Ordered Integral Domains / 199
4 Polynomials 202
4.1 Polynomials / 203
4.2 Factorization of Polynomials Over a Field / 214
4.3 Factor Rings of Polynomials Over a Field / 227
4.4 Partial Fractions / 236
4.5 Symmetric Polynomials / 239
4.6 Formal Construction of Polynomials / 248
5 Factorization in Integral Domains 251
5.1 Irreducibles and Unique Factorization / 252
5.2 Principal Ideal Domains / 264
6 Fields 274
6.1 Vector Spaces / 275
6.2 Algebraic Extensions / 283
6.3 Splitting Fields / 291
6.4 Finite Fields / 298
6.5 Geometric Constructions / 304
6.6 The Fundamental Theorem of Algebra / 308
6.7 An Application to Cyclic and BCH Codes / 310
7 Modules over Principal Ideal Domains 324
7.1 Modules / 324
7.2 Modules Over a PID / 335
8 p-Groups and the Sylow Theorems 349
8.1 Products and Factors / 350
8.2 Cauchy’s Theorem / 357
8.3 Group Actions / 364
8.4 The Sylow Theorems / 371
8.5 Semidirect Products / 379
8.6 An Application to Combinatorics / 382
9 Series of Subgroups 388
9.1 The Jordan-H¨older Theorem / 389
9.2 Solvable Groups / 395
9.3 Nilpotent Groups / 401
10 Galois Theory 412
10.1 Galois Groups and Separability / 413
10.2 The Main Theorem of Galois Theory / 422
10.3 Insolvability of Polynomials / 434
10.4 Cyclotomic Polynomials and Wedderburn’s Theorem /
442
11 Finiteness Conditions for Rings and Modules 447
11.1 Wedderburn’s Theorem / 448
11.2 The Wedderburn-Artin Theorem / 457
Appendices 471
Appendix A Complex Numbers / 471
Appendix B Matrix Algebra / 478
Appendix C Zorn’s Lemma / 486
Appendix D Proof of the Recursion Theorem / 490
BIBLIOGRAPHY 492
SELECTED ANSWERS 495
INDEX 523
关于作者
W. KEITH NICHOLSON, Ph D, 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.
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