Table of Contents
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