An Introduction to Essential Algebraic Structures

A reader–friendly introduction to modern algebra with important examples from various areas of mathematics Featuring a clear and concise approach , An Introduction to Essential Algebraic Structures presents an integrated approach to basic concepts of modern algebra and highlights topics that play a central role in various branches of mathematics. The authors discuss key topics of abstract and modern algebra including sets, number systems, groups, rings, and fields. The book begins with an exposition of the elements of set theory and moves on to cover the main ideas and branches of abstract algebra. In addition, the book includes: Numerous examples throughout to deepen readers’ knowledge of the presented material An exercise set after each chapter section in an effort to build a deeper understanding of the subject and improve knowledge retention Hints and answers to select exercises at the end of the book A supplementary website with an Instructors Solutions manual An Introduction to Essential Algebraic Structures is an excellent textbook for introductory courses in abstract algebra as well as an ideal reference for anyone who would like to be more familiar with the basic topics of abstract algebra. Martyn R. Dixon, PhD, is Professor in the Department of Mathematics at the University of Alabama. Dr. Dixon is the author of over 70 journal articles and two books, including Algebra and Number Theory: An Integrated Approach , also by Wiley. Leonid A. Kurdachenko, PhD, is Distinguished Professor and Chair of the Department of Algebra at the University of Dnepropetrovsk, Ukraine. Dr. Kurdachenko has authored over 200 journal articles as well as six books, including Algebra and Number Theory: An Integrated Approach , also by Wiley. Igor Ya. Subbotin, PhD , is Professor in the Department of Mathematics and Natural Sciences at National University in Los Angeles, California. Dr. Subbotin is the author of over 100 journal articles and six books, including Algebra and Number Theory: An Integrated Approach , also by Wiley.

Preface 3 Chapter 1. Sets 7 1.1 Operations on Sets 7 1.2 Set Mappings 15 1.3 Products of Mappings and Permutations 22 1.4 Operations on Matrices 34 1.5 Binary Algebraic Operations and Equivalence Relations 42 Chapter 2. Numbers 55 2.1 Some Properties of Integers. Mathematical Induction 55 2.2 Divisibility 60 2.3 Prime Factorization: The Fundamental Theorem of Arithmetic 67 2.4 Rational Numbers, Irrational Numbers and Real Numbers 72 Chapter 3. Groups 81 3.1 Groups and Subgroups 81 3.2 Cosets and Normal Subgroups 95 3.3 Factor Groups and Homomorphisms 109 Chapter 4. Rings 119 4.1 Rings, Subrings, Associative Rings 119   4.2 Rings of Polynomials 132   4.3 Ideals and Quotient Rings 142   4.4 Homomorphisms of Rings 154   Chapter 5. Fields 165 5.1 Fields: Basic Properties and Examples 165   5.2 Some Field Extensions 177   5.3 Fields of Algebraic Numbers 183   Hints and Answers to Selected Exercises 193 Index 217

Martyn R. Dixon, PhD, is Professor in the Department of Mathematics at the University of Alabama. Dr. Dixon is the author of over 70 journal articles and two books, including Algebra and Number Theory: An Integrated Approach , also by Wiley. Leonid A. Kurdachenko, PhD, is Distinguished Professor and Chair of the Department of Algebra at the University of Dnepropetrovsk, Ukraine. Dr. Kurdachenko has authored over 200 journal articles as well as six books, including Algebra and Number Theory: An Integrated Approach , also by Wiley. Igor Ya. Subbotin, PhD , is Professor in the Department of Mathematics and Natural Sciences at National University in Los Angeles, California. Dr. Subbotin is the author of over 100 journal articles and six books, including Algebra and Number Theory: An Integrated Approach , also by Wiley.