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Contemporary Abstract Algebra, Tenth Edition For more than three decades, this classic text has been widely appreciated by instructors and students alike. The book offers an enjoyable read and conveys and develops enthusiasm for the beauty of the topics presented. It is comprehensive, lively, and engaging. The author presents the concepts and methodologies of contemporary abstract algebra as used by working mathematicians, computer scientists, physicists, and chemists. Students will learn how to do computations and to write proofs. A unique feature of the book are exercises that build the skill of generalizing, a skill that students should develop but rarely do. Applications are included to illustrate the utility of the abstract concepts. Examples and exercises are the heart of the book. Examples elucidate the definitions, theorems, and proof techniques; exercises facilitate understanding, provide insight, and develop the ability to do proofs. The exercises often foreshadow definitions, concepts, and theorems to come. Changes for the tenth edition include new exercises, new examples, new quotes, and a freshening of the discussion portions. The hallmark features of previous editions of the book are enhanced in this edition. These include: A good mixture of approximately 1900 computational and theoretical exercises, including computer exercises, that synthesize concepts from multiple chapters Approximately 300 worked-out examples from routine computations to the challenging Many applications from scientific and computing fields and everyday life Historical notes and biographies that spotlight people and events Motivational and humorous quotations Numerous connections to number theory and geometry While many partial solutions and sketches for the odd-numbered exercises appear in the book, an Instructor's Solutions Manual written by the author has comprehensive solutions for all exercises and some alternative solutions to develop a critical thought and deeper understanding. It is available from CRC Press only. The Student Solution Manual has comprehensive solutions for all odd-numbered exercises and many even-numbered exercises. Author Joseph A. Gallian earned his PhD from Notre Dame. In addition to receiving numerous national awards for his teaching and exposition, he has served terms as the Second Vice President, and the President of the MAA. He has served on 40 national committees, chairing ten of them. He has published over 100 articles and authored six books. Numerous articles about his work have appeared in the national news outlets, including the New York Times, the Washington Post, the Boston Globe, and Newsweek, among many others Cover Half Title Series Page Title Page Copyright Page Dedication Contents Notations Preface 0. Preliminaries Properties of Integers Modular Arithmetic Complex Numbers Mathematical Induction Equivalence Relations Functions (Mappings) Exercises 1. Introduction to Groups Symmetries of a Square The Dihedral Groups Bibliography of Niels Abel 2. Groups Definition and Examples of Groups Elementary Properties of Groups Historical Note Exercises 3. Finite Groups; Subgroups Terminology and Notation Subgroup Tests Examples of Subgroups Exercises 4. Cyclic Groups Properties of Cyclic Groups Classification of Subgroups of Cyclic Groups Bibliography of James Joseph Sylvester 5. Permutation Groups Definitions and Notation Cycle Notation Properties of Permutations A Check-Digit Scheme Based on D5 Bibliography of Augustin Cauchy Bibliography of Alan Turing 6. Isomorphisms Motivation Definition and Examples Properties of Isomorphisms Automorphisms Cayley’s Theorem Exercises Bibliography of Arthur Cayley 7. Cosets and Lagrange’s Theorem Properties of Cosets Lagrange’s Theorem and Consequences An Application of Cosets to Permutation Groups The Rotation Group of a Cube and a Soccer Ball An Application of Cosets to the Rubik’s Cube Exercises Bibliography of Joseph Lagrange 8. External Direct Products Definition and Examples Properties of External Direct Products The Group of Units Modulo n as an External Direct Product Applications Exercises Bibliography of Leonard Adleman 9. Normal Subgroups and Factor Groups Normal Subgroups Factor Groups Applications of Factor Groups Internal Direct Products Exercises Bibliography of Évariste Galois 10. Group Homomorphisms Definition and Examples Properties of Homomorphisms The First Isomorphism Theorem Exercises Bibliography of Camille Jordan 11. Fundamental Theorem of Finite Abelian Groups The Fundamental Theorem The Isomorphism Classes of Abelian Groups Proof of the Fundamental Theorem Exercises 12. Introduction to Rings Motivation and Definition Examples of Rings Properties of Rings Subrings Exercises Bibliography of I. N. Herstein 13. Integral Domains Definition and Examples Fields Characteristic of a Ring Exercises 14. Ideals and Factor Rings Ideals Factor Rings Prime Ideals and Maximal Ideals Exercises Bibliography of Richard Dedekind Bibliography of Emmy Noether 15. Ring Homomorphisms Definition and Examples Properties of Ring Homomorphisms The Field of Quotients Exercises 16. Polynomial Rings Notation and Terminology The Division Algorithm and Consequences Exercises 17. Factorization of Polynomials Reducibility Tests Irreducibility Tests Unique Factorization in Z[x] Weird Dice: An Application of Unique Factorization Exercises Bibliography of Serge Lang 18. Divisibility in Integral Domains Irreducibles, Primes Historical Discussion of Fermat’s Last Theorem Unique Factorization Domains Euclidean Domains Exercises Bibliography of Sophie Germain Bibliography of Andrew Wiles Bibliography of Pierre de Fermat 19. Extension Fields The Fundamental Theorem of Field Theory Splitting Fields Zeros of an Irreducible Polynomial Exercises Bibliography of Leopold Kronecker 20. Algebraic Extensions Characterization of Extensions Finite Extensions Properties of Algebraic Extensions Exercises Bibliography of Ernst Steinitz 21. Finite Fields Classification of Finite Fields Structure of Finite Fields Subfields of a Finite Field Exercises Bibliography of L. E. Dickson Bibliography of E. H. Moore 22. Geometric Constructions Historical Discussion of Geometric Constructions Constructible Numbers Angle-Trisectors and Circle-Squarers Exercises 23. Sylow Theorems Conjugacy Classes The Class Equation The Sylow Theorems Applications of Sylow Theorems Exercises Bibliography of Ludwig Sylow 24. Finite Simple Groups Historical Background Nonsimplicity Tests The Simplicity of A5 The Fields Medal The Cole Prize Exercises Bibliography of Michael Aschbacher Bibliography of Daniel Gorenstein Bibliography of John Thompson 25. Generators and Relations Motivation Definitions and Notation Free Group Generators and Relations Classification of Groups of Order Up to 15 Characterization of Dihedral Groups Exercises Bibliography of Marshall Hall, Jr. 26. Symmetry Groups Isometries Classification of Finite Plane Symmetry Groups Classification of Finite Groups of Rotations in R3 Exercises 27. Symmetry and Counting Motivation Burnside’s Theorem Applications Group Action Exercises Bibliography of William Burnside 28. Cayley Digraphs of Groups Motivation The Cayley Digraph of a Group Hamiltonian Circuits and Paths Some Applications Exercises Bibliography of William Rowan Hamilton Bibliography of Paul Erdos 29. Introduction to Algebraic Coding Theory Motivation Linear Codes Parity-Check Matrix Decoding Coset Decoding Historical Note Exercises Bibliography of Richard W. Hamming Bibliography of Jessie MacWilliams Bibliography of Vera Pless 30. An Introduction to Galois Theory Fundamental Theorem of Galois Theory Solvability of Polynomials by Radicals Insolvability of a Quintic Exercises 31. Cyclotomic Extensions Motivation Cyclotomic Polynomials The Constructible Regular n-gons Exercises Bibliography of Carl Friedrich Gauss Bibliography of Manjul Bhargava Selected Answers Index