An Invitation to Abstract Algebra

Description

Studying abstract algebra can be an adventure of awe-inspiring discovery. The subject need not be watered down nor should it be presented as if all students will become mathematics instructors. This is a beautiful, profound, and useful field which is part of the shared language of many areas both within and outside of mathematics. To begin this journey of discovery, some experience with mathematical reasoning is beneficial. This text takes a fairly rigorous approach to its subject, and expects the reader to understand and create proofs as well as examples throughout. The book follows a single arc, starting from humble beginnings with arithmetic and high-school algebra, gradually introducing abstract structures and concepts, and culminating with Niels Henrik Abel and Evariste Galois’ achievement in understanding how we can―and cannot―represent the roots of polynomials. The mathematically experienced reader may recognize a bias toward commutative algebra and fondness for number theory. The presentation includes the following features: Exercises are designed to support and extend the material in the chapter, as well as prepare for the succeeding chapters. The text can be used for a one, two, or three-term course. Each new topic is motivated with a question. A collection of projects appears in Chapter 23. Abstract algebra is indeed a deep subject; it can transform not only the way one thinks about mathematics, but the way that one thinks―period. This book is offered as a manual to a new way of thinking. The author’s aim is to instill the desire to understand the material, to encourage more discovery, and to develop an appreciation of the subject for its own sake. Cover Half Title Series Page Title Page Copyright Page Dedication Contents Preface Author Symbols 1. Review of Sets, Functions, and Proofs 1.1. Sets 1.1.1. Some Special Sets of Numbers 1.1.2. Describing a Set 1.1.3. Operations on Sets 1.2. Functions 1.3. Proofs 1.3.1. Logic 1.3.2. Proof Conventions 1.4. How to Read This Book 2. Introduction: A Number Game 2.1. A Game with Integers 2.2. A Bigger Game 2.3. Concluding Remarks 2.4. Exercises 3. Groups 3.1. Introduction 3.2. Binary Operations 3.3. Groups: Definition and Some Examples 3.4. First Results about Groups 3.5. Exercises 4. Subgroups 4.1. Groups Inside Groups 4.2. The Subgroup Generated by a Set 4.3. Exercises 5. Symmetry 5.1. What is Symmetry? 5.2. Dihedral Groups 5.3. Exercises 6. Free Groups 6.1. The Free Group Generated by a Set 6.2. Exercises 7. Group Homomorphisms 7.1. Relationships between Groups 7.2. Kernels: How Much Did We Lose? 7.3. Cosets 7.4. Quotient Groups 7.5. Exercises 8. Lagrange’s Theorem 8.1. Cosets and Partitions 8.2. The Size of Cosets 8.3. Reaping the Consequences 8.4. Exercises 9. Special Types of Homomorphisms 9.1. Isomorphisms 9.2. Automorphisms 9.3. Embeddings 9.4. Exercises 10. Making Groups 10.1. Introduction 10.2. A Quotient Engine 10.3. Room for Everyone Inside 10.4. Exercises 11. Rings 11.1. A New Type of Structure 11.2. Ring Fundamentals 11.3. Ring Homomorphisms, Ideals, and Quotient Rings 11.4. Exercises 12. Results on Commutative Rings 12.1. Introduction 12.2. Primes and Domains 12.3. The Ideal Generated by a Set 12.4. Fields and Maximal Ideals 12.5. Exercises 13. Vector Spaces 13.1. Introduction 13.2. Abstract Vector Spaces 13.3. Bases: Generalized Coordinate Systems 13.4. Exercises 14. Polynomial Rings 14.1. Polynomials Over a Commutative Ring 14.2. Polynomials Over a Field 14.3. Exercises 15. Field Theory 15.1. Extension Fields 15.2. Splitting Fields 15.3. Exercises 16. Galois Theory 16.1. Field Embeddings 16.2. Separable Extensions 16.3. Normal Extensions 16.4. Galois Extensions 16.5. Exercises 17. Direct Sums and Direct Products 17.1. Introduction 17.2. Direct Products 17.3. Direct Sums 17.4. Exercises 18. The Structure of Finite Abelian Groups 18.1. Introduction 18.2. Preliminaries 18.3. Splitting into p-Subgroups 18.4. Structure of Abelian p-Groups 18.5. The Fundamental Theorem 18.6. Exercises 19. Group Actions 19.1. Groups Acting on Sets 19.2. Reaping the Consequences 19.3. Exercises 20. Learning from Z 20.1. Introduction 20.2. Fractions 20.3. Unique Factorization 20.4. Exercises 21. The Problems of the Ancients 21.1. Introduction 21.2. Constructible Numbers 21.3. Constructible Regular Polygons 21.4. Exercises 22. Solvability of Polynomial Equations by Radicals 22.1. Radicals 22.2. Solvable Polynomials 22.3. Solvable Groups 22.4. Galois Groups in the Generic Case 22.5. Which Groups Are Solvable? 22.6. The Grand Finale 22.7. Exercises 23. Projects 23.1. Gyrogroups 23.2. Kaleidoscopes 23.3. The Axiom of Choice 23.4. Some Category Theory 23.5. Linear Algebra: Change of Basis 23.6. Linear Algebra: Determinants 23.7. Linear Algebra: Eigenvalues 23.8. Linear Algebra: Rotations 23.9. Power Series 23.10. Quadratic Probing 23.11. Euclidean Domains 23.12. Resultants 23.13. Perfect Numbers and Lucas’s Test 23.14. Modules Bibliography Index