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A First Course in Abstract Algebra, 8th Edition retains its hallmark goal of covering all the topics needed for an in-depth introduction to abstract algebra – and is designed to be relevant to future graduate students, future high school teachers, and students who intend to work in industry. New co-author Neal Brand has revised this classic text carefully and thoughtfully, drawing on years of experience teaching the course with this text to produce a meaningful and worthwhile update. This in-depth introduction gives students a firm foundation for more specialized work in algebra by including extensive explanations of the what, the how, and the why behind each method the authors choose. This revision also includes applied topics such as RSA encryption and coding theory, as well as examples of applying Gröbner bases. Key to the 8th Edition has been transforming from a print-based learning tool to a digital learning tool. The eText is packed with content and tools, such as mini-lecture videos and interactive figures, that bring course content to life for students in new ways and enhance instruction. A low-cost, loose-leaf version of the text is also available for purchase within the Pearson eText. Front Cover Title Page Copyright Page Contents Instructor’s Preface Dependence Chart Student’s Preface 0 Sets and Relations I Groups and Subgroups 1 Binary Operations 2 Groups 3 Abelian Examples 4 Nonabelian Examples 5 Subgroups 6 Cyclic Groups 7 Generating Sets and Cayley Digraphs II Structure of Groups 8 Groups of Permutations 9 Finitely Generated Abelian Groups 10 Cosets and the Theorem of Lagrange 11 Plane Isometries III Homomorphisms and Factor Groups 12 Factor Groups 13 Factor-Group Computations and Simple Groups 14 Group Action on a Set 15 Applications of G-Sets to Counting IV Advanced Group Theory 16 Isomorphism Theorems 17 Sylow Theorems 18 Series of Groups 19 Free Abelian Groups 20 Free Groups 21 Group Presentations V Rings and Fields 22 Rings and Fields 23 Integral Domains 24 Fermat’s and Euler’s Theorems 25 Encryption VI Constructing Rings and Fields 26 the Field of Quotients of an Integral Domain 27 Rings of Polynomials 28 Factorization of Polynomials over a Field 29 Algebraic Coding Theory 30 Homomorphisms and Factor Rings 31 Prime and Maximal Ideals 32 Noncommutative Examples VII Commutative Algebra 33 Vector Spaces 34 Unique Factorization Domains 35 Euclidean Domains 36 Number Theory 37 Algebraic Geometry 38 Gröbner Bases for Ideals VIII Extension Fields 39 Introduction to Extension Fields 40 Algebraic Extensions 41 Geometric Constructions 42 Finite Fields IX Galois Theory 43 Introduction to Galois Theory 44 Splitting Fields 45 Separable Extensions 46 Galois Theory 47 Illustrations of Galois Theory 48 Cyclotomic Extensions 49 Insolvability of the Quintic Appendix: Matrix Algebra Bibliography Notations Answers to Odd-numbered Exercises Not Asking for Definitions or Proofs Index Main subject categories: • Algebra • Groups and subgroups • Structure of groups • Homomorphisms and factor groups • Rings • Fields • Constructing rings and fields • Commutative algebra • Extension fields • Galois theoryA First Course in Abstract Algebra, 8th Edition retains its hallmark goal of covering all the topics needed for an in-depth introduction to abstract algebra – and is designed to be relevant to future graduate students, future high school teachers, and students who intend to work in industry. New co-author Neal Brand has revised this classic text carefully and thoughtfully, drawing on years of experience teaching the course with this text to produce a meaningful and worthwhile update. This in-depth introduction gives students a firm foundation for more specialized work in algebra by including extensive explanations of the what, the how, and the why behind each method the authors choose. This revision also includes applied topics such as RSA encryption and coding theory, as well as examples of applying Gröbner bases. "This is an introduction to abstract algebra. It is anticipated that the students have studied calculus and probably linear algebra. However, these are primarily mathematical maturity prerequisites; subject matter from calculus and linear algebra appears mostly in illustrative examples and exercises. As in previous editions of the text, my aim remains to teach students as much about groups, rings, and fields as I can in a first course. For many students, abstract algebra is their first extended exposure to an axiomatic treatment of mathematics. Recognizing this, I have included extensive explanations concerning what we are trying to accomplish, how we are trying to do it, and why we choose these methods. Mastery of this text constitutes a firm foundation for more specialized work in algebra, and also provides valuable experience for any further axiomatic study of mathematics"-- Provided by publisher