Read more
Informationen zum Autor 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. Klappentext Praise for the Third Edition". . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . ."--Zentralblatt MATHThe 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 theoremThroughout 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. Zusammenfassung This is a self-contained introduction to the basic structures of abstract algebra and its applications. Classroom-tested over several decades, the book is self-contained and is ideal for self-study. The author has thoroughly reviewed and revised the book and has also significantly added to the discussion on modules over principle ideal domains. Inhaltsverzeichnis 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 Fie...
