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Praise for the Third Edition &quote;. . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . .&quote; 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…mehr

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Produktbeschreibung
Praise for the Third Edition "e;. . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . ."e; 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.

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  • Produktdetails
  • Verlag: John Wiley & Sons
  • Seitenzahl: 560
  • Erscheinungstermin: 23. Februar 2012
  • Englisch
  • ISBN-13: 9781118311738
  • Artikelnr.: 38260092
Autorenporträt
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.
Inhaltsangabe
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