## 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.

Preface
To The Instructor
To The Student

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*

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
10.4 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

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