Introduction: Prerequisites and Preliminaries

• 1 Logic 1
• 2 Sets and Classes
• 3 Functions
• 4 Relations and Partitions
• 5 Products
• 6 The Integers
• 7 The Axiom of Choice, Order and Zorn's Lemma
• 8 Cardinal Numbers

Chapter 1: Groups

• 1 Semigroups, Monoids and Groups
• 2 Homomorphisms and Subgroups.
• 3 Cyclic Groups
• 4 Cosets and Counting
• 5 Normality, Quotient Groups, and Homomorphisms
• 6 Symmetric, Alternating, and Dihedral Groups
• 7 Categories: Products, Coproducts, and Free Objects
• 8 Direct Products and Direct Sums
• 9 Free Groups, Free Products, Generators & Relations

Chapter II: The Structure of Groups

• 1 Free Abelian Groups
• 2 Finitely Generated Abelian Groups
• 3 The Krull-Schmidt Theorem
• 4 The Action of a Group on a Set
• 5 The Sylow Theorems
• 6 Classification of Finite Groups
• 7 Nilpotent and Solvable Groups
• 8 Normal and Subnormal Series

Chapter Ill: Rings

• 1 Rings and Homomorphisms
• 2 Ideals
• 3 Factorization in Commutative Rings
• 4 Rings of Quotients and Localization
• 5 Rings of Polynomials and Formal Power Series
• 6 Factorization in Polynomial Rings.

Chapter IV: Modules

• 1 Modules, Homomorphisms and Exact Sequences
• 2 Free Modules and Vector Spaces
• 3 Projective and Injective Modules
• 4 Hom and Duality
• 5 Tensor Products
• 6 Modules over a Principal Ideal Domain
• 7 Algebras

Chapter V: Fields and Galois Theory

• 1 Field Extensions
  • Appendix: Ruler and Compass Constructions
• 2 The Fundamental Theorem
  • Appendix: Symmetric Rational Functions