Mark R. Sepanski, Baylor University, Waco, TX

Preface
Chapter 1. Arithmetic
1. Integers
1.1. Basic Properties
1.2. Induction
1.3. Division Algorithm
1.4. Divisors
1.5. Fundamental Theorem of Arithmetic
1.6. Exercises 1.1–1.28

2. Modular Arithmetic
2.1. Congruence
2.2. Congruence Classes
2.3. Arithmetic
2.4. Structure
2.5. Applications
2.6. Equivalence Relations
2.7. Exercises 1.29–1.72

Chapter 2. Groups
1. Definitions and Examples
1.1. Binary Operations
1.2. Definition of a Group
1.3. Examples
1.4. Exercises 2.1–2.29
2. Basic Properties and Order
2.1. Exercises 2.30–2.52

3. Subgroups and Direct Products
3.1. Subgroups
3.2. Direct Products
3.3. Exercises 2.53–2.91

4. Morphisms
4.1. Introduction
4.2. Definitions and Examples
4.3. Basic Properties
4.4. Exercises 2.92–2.118

5. Quotients
5.1. Definitions
5.2. Lagrange's Theorem
5.3. Normality
5.4. Direct Products and the Correspondence Theorem
5.5. First Isomorphism Theorem
5.6. Exercises 2.119–2.169

6. Fundamental Theorem of Finite Abelian Groups
6.1. Exercises 2.170–2.188

7. The Symmetric Group
7.1. Cayley's Theorem
7.2. Cyclic Decomposition
7.3. Conjugacy Classes
7.4. Parity and the Alternating Subgroup
7.5. Exercises 2.189–2.229

8. Group Actions
8.1. Exercises 2.230–2.248

9. Sylow Theorems
9.1. Exercises 2.249–2.281

10. Simple Groups and Composition Series
10.1. Simple Groups
10.2. Composition Series
10.3. Exercises 2.282–2.301

Chapter 3. Rings
1. Examples and Basic Properties
1.1. Definition
1.2. Examples
1.3. Basic Properties
1.4. Subrings
1.5. Direct Products
1.6. Exercises 3.1–3.39

2. Morphisms and Quotients
2.1. Morphisms
2.2. Ideals
2.3. Quotients
2.4. Isomorphism Theorem
2.5. Exercises 3.40–3.86

3. Polynomials and Roots
3.1. Division Algorithm
3.2. Roots
3.3. Rational Root Test
3.4. Fundamental Theorem of Algebra
3.5. Exercises 3.87–3.129

4. Polynomials and Irreducibility
4.1. Irreducibility
4.2. Polynomials over a Field
4.3. Exercises 3.130–3.150

5. Factorization
5.1. Unique Factorization
5.2. Quotient Fields
5.3. Unique Factorization in Polynomial Rings
5.4. Exercises 3.151–3.172

6. Principal Ideal and Euclidean Domains
6.1. Principal Ideal Domains
6.2. Euclidean Domains
6.3. Gaussian Integers
6.4. Exercises 3.173–3.201

Chapter 4. Field Theory
1. Finite and Algebraic Extensions
1.1. Vector Spaces
1.2. Finite and Algebraic Extensions
1.3. Exercises 4.1–4.34

2. Splitting Fields
2.1. Splitting Fields
2.2. Algebraic Closures
2.3. Exercises 4.35–4.62

3. Finite Fields
3.1. Exercises 4.63–4.81

4. Galois Theory
4.1. Galois Groups
4.2. Separability
4.3. Galois Correspondence
4.4. Exercises 4.82—4.135

5. Famous Impossibilities
5.1. Compass and Straightedge Constructions
5.2. Solvability of Polynomials
5.3. Exercises 4.136–4.162

6. Cyclotomic Fields
6.1. Exercises 4.163–4.179
Index