**Abstract Algebra: An Introduction By Thomas Hungerford**

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.

**Book Description:**

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.

**[PDF] Abstract Algebra: An Introduction Table Of Contents**

Preface

To The Instructor

To The Student

Thematic Table of Contents for the Core Course

Part 1 The Core Course

CHAPTER 1 Arithmetic in Z Revisited

1.1 The Division Algorithm

1.2 Divisibility

1.3 Primes and Unique Factorization

CHAPTER 2 Conuruence inZ and Modular Arithmetic

2.1 Congruence and Congruence Classes

2.2 Modular Arithmetic

2.3 The Structure of ZP (p Prime) and Zn

CHAPTER 3 Rings

3.1 Definition and Examples of Rings

3.2 Basic Properties of Rings

3.3 Isomorphisms and Homomorphisms

CHAPTER 4 Arithmetic in f[x]
4.1 Polynomial Arithmetic and the Division Algorithm

4.2 Divisibility in F[x]
4.3 lrreducibles and Unique Factorization

4.4 Polynomial Functions, Roots, and Reducibility

4.5 Irreducibility in Q[x]*

4.6 Irreducibility in R[x] and C[x]*

CHAPTER 5 Congruence in f[x] and Congruence-Glass Arithmetic

5.1 Congruence in F[x] and Congruence Classes

5.2 Congruence-Class Arithmetic

5.3 The Structure of F[x]/(p(x)) When p(x) Is Irreducible

CHAPTER 6 Ideals and Quotient Rings

6.1 Ideals and Congruence

6.2 Quotient Rings and Homomorphisms

6.3 The Structure of R//When /Is Prime or Maximal*

CHAPTER 7 Groups

7.1 Definition and Examples of Groups

7.2 Basic Properties of Groups

7.3 Subgroups

7.4 Isomorphisms and Homomorphisms*

7.5 The Symmetric and Alternating Groups*

CHAPTER 8 Normal Subgroups and Quotient Groups

8.1 Congruence and Lagrange’s Theorem

8.2 Normal Subgroups

8.3 Quotient Groups

8.4 Quotient Groups and Homomorphisms

II The Simplicity of An*

Part 2 Advanced Topics

CHAPTER 9 Topics in Group Theory

9.1 Direct Products

9.2 Finite Abelian Groups

9.3 The Sylow Theorems

9.4 Conjugacy and the Proof of the Sylow Theorems

9.5 The Structure of Finite Groups

CHAPTER 10 Arithmetic in Integral Domains

10.1 Euclidean Domains

10.2 Principal Ideal Domains and Unique FactorizationDomains

10.3 Factorization of Quadratic Integers*

10.4 The Field of Quotients of an Integral Domain*

10.5 Unique Factorization in Polynomial Domains*

CHAPTER 11 Field Extensions

11.1 Vector Spaces

11.2 Simple Extensions

11.3 Algebraic Extensions

11.4 Splitting Fields

11.5 Separability

11.6 Finite Fields

CHAPTER 12 Galois Theory

12.1 The Galois Group

12.2 The Fundamental Theorem of Galois Theory

12.3 Solvability by Radicals

Part 3 Excursions and Applications

CHAPTER 13 Public-Key Cryptography

CHAPTER 14 The Chinese Remainder Theorem

14.1 Proof of the Chinese Remainder Theorem

14.2 Applications of the Chinese Remainder Theorem

14.3 The Chinese Remainder Theorem for Rings

CHAPTER 15 Geometric Constructions

CHAPTER 16 Algebraic Coding Theory

16.1 Linear Codes

16.2 Decoding Techniques

16.3 BCH Codes

Part 4 Appendices

A. Logic and Proof

B. Sets and Functions

C. Well Ordering and Induction

D. Equivalence Relations

E. The Binomial Theorem

F. Matrix Algebra

6. Polynomials

Bibliography

Answers and Suggestions for Selected Odd-Numbered

Index

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