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.### 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

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.

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

- Author: Thomas W. Hungerford
- Edition: 3rd
- Publication Date: 2014
- Publisher: Cengage Learning
- ISBN-13: 9781111569624
- Pages: 621
- Format: pdf
- Size: 63.5M

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