Abstract Algebra

Title
Preface
Contents
Chapter 1. Preliminaries
1.1. Set Operations and Functions
1.2. Partitions and Equivalence Relations
1.3. Binary Operations
1.4. The Integers
Chapter 2. Groups
2.1. Groups and Subgroups
2.2. Homomorphisms
2.3. Lagrange's Theorem and the Quotient Set
2.4. Normal Subgroups and the Quotient Group
Chapter 3. Group Actions and Solvable Groups
3.1. Group Actions
3.2. Solvable Groups
Chapter 4. Rings
4.1. Rings and Subrings
4.2. Homomorphisms, Ideals, and Quotient Rings
4.3. Maximal Ideals and the Chinese Remainder Theorem
4.4. Prime Ideals, Integral Domains, and the Fraction Field
Chapter 5. Factorization in Commutative Rings
5.1. Euclidean Rings and Principal Ideal Rings
5.2. Primes and Unique Factorization
5.3. Factorization and Noetherian Domains
Chapter 6. Algebras
6.1. Algebras and Morphisms
6.2. Polynomials
6.3. Matrices and Determinants
Chapter 7. Modules and Vector Spaces
7.1. Left R-modules
7.2. Direct Products and Direct Sums
7.3. Vector Spaces
Chapter 8. Field Extensions
8.1. Finitely Generated Extensions
8.2. Algebraic Extensions
8.3. Splitting Fields, Normal Extensions, and Finite Fields
Chapter 9. Galois Theory
9.1. The Fundamental Correspondence
9.2. The Fundamental Theorem of Galois Theory
9.3. Solvability by Radicals
Appendix A. Zorn's Lemma
Appendix B. Categories and Functors
Bibliography
Index