Chapter 1: Preliminaries
1.1 Sets
1.2 Relations
1.3 Functions
1.4 Natural Numbers
1.5 Integers
1.6 Counting
1.7 Partially Ordered Sets
Chapter 2: Elementary Group Theory
2.1 Definition of a Group and Examples
2.2 Basic Properties of Groups
2.3 Subgroups
2.4 Cyclic Groups
2.5 The Symmetric Group
2.6 Products of Subgroups
2.7 Normal Subgroups
2.8 Quotient Groups
2.9 Homomorphisms
2.10 Automorphisms of Groups
2.11 Alternating Groups
2.12 Group Actions
2.13 The Class Equation
2.14 Sylow’s Theorems
2.15 Direct Products of Groups
2.16 Finite Abelian Groups
Chapter 3: Rings
3.1 Introduction to Rings
3.2 Integral Domains
3.3 Polynomial Rings
3.4 Homomorphisms and Ideals
3.5 Quotient Rings
3.6 Prime and Maximal Ideals in Commutative Rings
3.7 Field of Fractions of an Integral Domain
3.8 Principal Ideal Domains and Euclidean Domains
3.9 Polynomials Over Fields
3.10 The Gaussian Integers
3.11 Unique Factorization Domains
Chapter 4: Vector Spaces
4.1 Vector Spaces Over a Field F
4.2 Span and Independence
4.3 Bases and Dimension
Chapter 5: Composition Series and Solvable Groups
5.1 Compositions Series and the Jordan-Holder Theorem
5.2 Solvable Groups
Chapter 6: Fields
6.1 Extensions of Fields
6.2 Splitting Fields and Roots of Polynomials
6.3 Finite Fields
6.4 Constructible Numbers
6.5 Galois Theory
6.6 Cyclotomic Polynomials and Extensions
6.7 Solvability by Radicals
6.8 Cubic and Quartic Polynomials