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.
Author(s): Thomas W. Hungerford
Edition: 3rd
Publisher: Brooks/Cole, Cengage Learning
Year: 2014
Language: English
Commentary: Enhanced
Pages: C, xvii, 595
City: Boston
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
