Abstract Algebra: An Integrated Approach

Preface
Chapter 1. A Potpourri of Preliminary Topics
	1.1. What Are Definitions, Axioms, and Proofs?
	1.2. Mathematical Credos to Live By!
	1.3. A Smidgeon of Mathematical Logic and Some Proof Techniques
	1.4. A Smidgeon of Set Theory
	1.5. Functions
	1.6. Equivalence Relations
	1.7. Mathematical Induction
	1.8. A Smidgeon of Number Theory
	1.9. A Smidgeon of Combinatorics
	Exercises
Chapter 2. Groups — Part 1
	2.1. Introduction to Groups
	2.2. Abstract Groups
	2.3. Interesting Examples of Groups
	2.4. Group Homomorphisms
	2.5. Subgroups, Cosets, and Lagrange's Theorem
	2.6. Products of Groups
	Exercises
Chapter 3. Rings — Part 1
	3.1. Introduction to Rings
	3.2. Abstract Rings and Ring Homomorphisms
	3.3. Interesting Examples of Rings
	3.4. Some Important Special Types of Rings
	3.5. Unit Groups and Product Rings
	3.6. Ideals and Quotient Rings
	3.7. Prime Ideals and Maximal Ideals
	Exercises
Chapter 4. Vector Spaces — Part 1
	4.1. Introduction to Vector Spaces
	4.2. Vector Spaces and Linear Transformations
	4.3. Interesting Examples of Vector Spaces
	4.4. Bases and Dimension
	Exercises
Chapter 5. Fields — Part 1
	5.1. Introduction to Fields
	5.2. Abstract Fields and Homomorphisms
	5.3. Interesting Examples of Fields
	5.4. Subfields and Extension Fields
	5.5. Polynomial Rings
	5.6. Building Extension Fields
	5.7. Finite Fields
	Exercises
Chapter 6. Groups — Part 2
	6.1. Normal Subgroups and Quotient Groups
	6.2. Groups Acting on Sets
	6.3. The Orbit-Stabilizer Counting Theorem
	6.4. Sylow's Theorem
	6.5. Two Counting Lemmas
	6.6. Double Cosets and Sylow's Theorem
	Exercises
Chapter 7. Rings — Part 2
	7.1. Irreducible Elements and Unique Factorization Domains
	7.2. Euclidean Domains and Principal Ideal Domains
	7.3. Factorization in Principal Ideal Domains
	7.4. The Chinese Remainder Theorem
	7.5. Field of Fractions
	7.6. Multivariate and Symmetric Polynomials
	Exercises
Chapter 8. Fields — Part 2
	8.1. Algebraic Numbers and Transcendental Numbers
	8.2. Polynomial Roots and Multiplicative Subgroups
	8.3. Splitting Fields, Separability, and Irreducibility
	8.4. Finite Fields Revisited
	8.5. Gauss's Lemma and Eisenstein's Irreducibility Criterion
	8.6. Ruler and Compass Constructions
	Exercises
Chapter 9. Galois Theory: Fields+Groups
	9.1. What Is Galois Theory?
	9.2. A Quick Review of Polynomials and Field Extensions
	9.3. Fields of Algebraic Numbers
	9.4. Algebraically Closed Fields
	9.5. Automorphisms of Fields
	9.6. Splitting Fields — Part 1
	9.7. Splitting Fields — Part 2
	9.8. The Primitive Element Theorem
	9.9. Galois Extensions
	9.10. The Fundamental Theorem of Galois Theory
	9.11. Application: The Fundamental Theorem of Algebra
	9.12. Galois Theory of Finite Fields
	9.13. A Plethora of Galois Equivalences
	9.14. Cyclotomic Fields and Kummer Fields
	9.15. Application: Insolubility of Polynomial Equations by Radicals
	9.16. Linear Independence of Field Automorphisms
	Exercises
Chapter 10. Vector Spaces — Part 2
	10.1. Vector Space Homomorphisms (aka Linear Transformations)
	10.2. Endomorphisms and Automorphisms
	10.3. Linear Transformations and Matrices
	10.4. Subspaces and Quotient Spaces
	10.5. Eigenvalues and Eigenvectors
	10.6. Determinants
	10.7. Determinants, Eigenvalues, and Characteristic Polynomials
	10.8. Inifinite-Dimensional Vector Spaces
	Exercises
Chapter 11. Modules — Part 1:Rings+Vector-Like Spaces
	11.1. What Is a Module?
	11.2. Examples of Modules
	11.3. Submodules and Quotient Modules
	11.4. Free Modules and Finitely Generated Modules
	11.5. Homomorphisms, Endomorphisms, Matrices
	11.6. Noetherian Rings and Modules
	11.7. Matrices with Entries in a Euclidean Domain
	11.8. Finitely Generated Modules over Euclidean Domains
	11.9. Applications of the Structure Theorem
	Exercises
Chapter 12. Groups — Part 3
	12.1. Permutation Groups
	12.2. Cayley's Theorem
	12.3. Simple Groups
	12.4. Composition Series
	12.5. Automorphism Groups
	12.6. Semidirect Products of Groups
	12.7. The Structure of Finite Abelian Groups
	Exercises
Chapter 13. Modules — Part 2: Multilinear Algebra
	13.1. Multilinear Maps and Multilinear Forms
	13.2. Symmetric and Alternating Forms
	13.3. Alternating Forms on Free Modules
	13.4. The Determinant Map
	Exercises
Chapter 14. Additional Topics in Brief
	14.1. Sets Countable and Uncountable
	14.2. The Axiom of Choice
	14.3. Tensor Products and Multilinear Algebra
	14.4. Commutative Algebra
	14.5. Category Theory
	14.6. Graph Theory
	14.7. Representation Theory
	14.8. Elliptic Curves
	14.9. Algebraic Number Theory
	14.10. Algebraic Geometry
	14.11. Euclidean Lattices
	14.12. Non-Commutative Rings
	14.13. Mathematical Cryptography
	Exercises
Sample Syllabi
List of Notation
List of Figures
Index