Introduction to Abstract Algebra

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. Contetns Chapter Dependency Diagram Notation used in the Text Chapter 0 Preliminaries 0.1 Proofs 0.2 Sets 0.3 Mappings 0.4 Equivalences Chapter 1 Integers and Permutations 1.1 Induction 1.2 Divisors and Prime Factorization 1.3 Integers Modulo n 1.4 Permutations Chapter 2 Groups 2.1 Binary Operations 2.2 Groups 2.3 Subgroups 2.4 Cyclic Groups and the Order of an Element 2.5 Homomorphisms and Isomorphisms 2.6 Cosets and Lagrange's Theorem 2.7 Groups of Motions and Symmetries 2.8 Normal Subgroups 2.9 Factor Groups 2.10 The Isomorphism Theorem 2.11 An Application to Binary Linear Codes Chapter 3 Rings 3.1 Examples and Basic Properties 3.2 Integral Domains and Fields 3.3 Ideals and Factor Rings 3.4 Homomorphisms 3.5 Ordered Integral Domains Chapter 4 Polynomials 4.1 Polynomials 4.2 Factorization of Polynomials over a Field 4.3 Factor Rings of Polynomials over a Field 4.4 Partial Fractions 4.5 Symmetric Polynomials 4.6 Formal Construction of Polynomials Chapter 5 Factorization in Integral Domains 5.1 Irreducibles and Unique Factorization 5.2 Principal Ideal Domains Chapter 6 Fields 6.1 Vector Spaces 6.2 Algebraic Extensions 6.3 Splitting Fields 6.4 Finite Fields 6.5 Geometric Constructions 6.6 The Fundamental Theorem of Algebra 6.7 An Application to Cyclic and BCH Codes Chapter 7 Modules over Principal Ideal Domains 7.1 Modules 7.2 Modules over a PID Chapter 8 p-Groups and the Sylow Theorems 8.1 Products and Factors 8.2 Cauchy's Theorem 8.3 Group Actions 8.4 The Sylow Theorems 8.5 Semidiect Products 8.6 An Application to Combinatorics Chapter 9 Series of Subgroups 9.1 The Jordan-Holder Theorem 9.2 Solvable Groups 9.3 Nilpotent Groups Chapter 10 Galois Theory 10.1 Galois Groups and Separability 10.2 The Main Theorem of Galois Theory 10.3 Insolvability of Polynomials 10.4 Cyclotomic Polynomials and Wedderburn's Theorem Chapter 11 Finiteness Conditions for Rings and Modules 11.1 Wedderburn's Theorem 11.2 The Wedderburn-Artin Theorem Appendices Appendix A Complex Numbers Appendix B Matrix Algebra Appendix C Zorn's Lemma Appendix D Proof of the Recursion Theorem Bibliography Selected Answers Index