ABSTRACT ALGEBRA: AN INTRODUCTION, 3E, It is intended for a first undergraduate course in modern abstract algebra. The flexible design of the text makes it suitable for courses of various lengths and different levels of mathematical sophistication, ranging from a traditional abstract algebra course to one with a more applied flavor. The emphasis is on clarity of exposition. The thematic development and organizational overview is what sets this book apart. The chapters are organized around three themes: arithmetic, congruence, and abstract structures. Each theme is developed first for the integers, then for polynomials, and finally for rings and groups. This enables students to see where many abstract concepts come from, why they are important, and how they relate to one another. Features The flexible design of the text makes it suitable for courses of various lengths and different levels of mathematical sophistication flavor. The emphasis of this text is on clarity of exposition. The chapters are organized around three themes: arithmetic, congruence, and abstract structures. The interconnections of the basic areas of algebra are frequently pointed out in the text and in the Thematic Table of Contents. Cover......Page 1 Notations......Page 2 Contents......Page 9 Preface......Page 13 To The Instructor......Page 16 To The Student......Page 18 Thematic Table of Contents for the Core Course......Page 20 Part 1 The Core Course......Page 23 1.1 The Division Algorithm......Page 25 1.2 Divisibility......Page 31 1.3 Primes and Unique Factorization......Page 39 2.1 Congruence and Congruence Classes......Page 47 2.2 Modular Arithmetic......Page 54 2.3 The Structure of ZP (p Prime) and Zn......Page 59 CHAPTER 3 Rings......Page 65 3.1 Definition and Examples of Rings......Page 66 3.2 Basic Properties of Rings......Page 81 3.3 Isomorphisms and Homomorphisms......Page 92 CHAPTER 4 Arithmetic in f[x]......Page 107 4.1 Polynomial Arithmetic and the Division Algorithm......Page 108 4.2 Divisibility in F[x]......Page 117 4.3 lrreducibles and Unique Factorization......Page 122 4.4 Polynomial Functions, Roots, and Reducibility......Page 127 4.5 Irreducibility in Q[x]*......Page 134 4.6 Irreducibility in R[x] and C[x]*......Page 142 5.1 Congruence in F[x] and Congruence Classes......Page 147 5.2 Congruence-Class Arithmetic......Page 152 5.3 The Structure of F[x]/(p(x)) When p(x) Is Irreducible......Page 157 6.1 Ideals and Congruence......Page 163 6.2 Quotient Rings and Homomorphisms......Page 174 6.3 The Structure of R//When /Is Prime or Maximal*......Page 184 7.1 Definition and Examples of Groups......Page 191 7.2 Basic Properties of Groups......Page 218 7.3 Subgroups......Page 225 7.4 Isomorphisms and Homomorphisms*......Page 236 7.5 The Symmetric and Alternating Groups*......Page 249 8.1 Congruence and Lagrange's Theorem......Page 259 8.2 Normal Subgroups......Page 270 8.3 Quotient Groups......Page 277 8.4 Quotient Groups and Homomorphisms......Page 285 II The Simplicity of An*......Page 295 Part 2 Advanced Topics......Page 301 9.1 Direct Products......Page 303 9.2 Finite Abelian Groups......Page 311 9.3 The Sylow Theorems......Page 320 9.4 Conjugacy and the Proof of the Sylow Theorems......Page 326 9.5 The Structure of Finite Groups......Page 334 CHAPTER 10 Arithmetic in Integral Domains......Page 343 10.1 Euclidean Domains......Page 344 10.2 Principal Ideal Domains and Unique FactorizationDomains......Page 354 10.3 Factorization of Quadratic Integers*......Page 366 10.4 The Field of Quotients of an Integral Domain*......Page 375 10.5 Unique Factorization in Polynomial Domains*......Page 381 11.1 Vector Spaces......Page 387 11.2 Simple Extensions......Page 398 11.3 Algebraic Extensions......Page 404 11.4 Splitting Fields......Page 410 11.5 Separability......Page 416 11.6 Finite Fields......Page 421 12.1 The Galois Group......Page 429 12.2 The Fundamental Theorem of Galois Theory......Page 437 12.3 Solvability by Radicals......Page 445 Part 3 Excursions and Applications......Page 457 CHAPTER 13 Public-Key Cryptography......Page 459 14.1 Proof of the Chinese Remainder Theorem......Page 465 14.2 Applications of the Chinese Remainder Theorem......Page 472 14.3 The Chinese Remainder Theorem for Rings......Page 475 CHAPTER 15 Geometric Constructions......Page 481 16.1 Linear Codes......Page 493 16.2 Decoding Techniques......Page 505 16.3 BCH Codes......Page 514 Part 4 Appendices......Page 521 A. Logic and Proof......Page 522 B. Sets and Functions......Page 531 C. Well Ordering and Induction......Page 545 D. Equivalence Relations......Page 553 E. The Binomial Theorem......Page 559 F. Matrix Algebra......Page 562 6. Polynomials......Page 567 Bibliography......Page 575 Answers and Suggestions for Selected Odd-Numbered......Page 578 Index......Page 611 ABSTRACT ALGEBRA: AN INTRODUCTION is intended for a first undergraduate course in modern abstract algebra. Its flexible design makes it suitable for courses of various lengths and different levels of mathematical sophistication, ranging from a traditional abstract algebra course to one with a more applied flavor. The book is organized around two themes: arithmetic and congruence. Each theme is developed first for the integers, then for polynomials, and finally for rings and groups, so students can see where many abstract concepts come from, why they are important, and how they relate to one another. New Features: - A groups-first option that enables those who want to cover groups before rings to do so easily. - Proofs for beginners in the early chapters, which are broken into steps, each of which is explained and proved in detail. - In the core course (chapters 1-8), there are 35% more examples and 13% more exercises.
Abstract Algebra: An Introduction is set apart by its thematic development and organization. The chapters are organized around two themes: arithmetic and congruence. Each theme is developed first for the integers, then for polynomials, and finally for rings and groups. This enables students to see where many abstract concepts come from, why they are important, and how they relate to one another. New to this edition is a "groups first" option that enables those who prefer to cover groups before rings to do so easily.
