The Basics of Abstract Algebra

Title
Contents
Preface
0. Preliminaries
	0.1. Sets
	0.2. Relations
	0.3. Functions
	0.4. Binary Operations
	0.5. Matrices
1. The Integers
	1.1. The Integers
	1.2. Mathematical induction
	1.3. Congruence relations and modular arithmetic
2. Sets with One Binary Operation: Groups
	2.1. Groups
	2.2. Subgroups and factor groups
	2.3. Algebraically equivalent groups: group homomorphisms
3. Sets with Two Binary Operations: Rings
	3.1. Rings
	3.2. Subrings, ideals, and factor rings
	3.3. Rings that are algebraically equivalent: ring homomorphisms
4. The Rational, Real, and Complex Number Systems
	4.1. Fields of fractions and the rational numbers
	4.2. Ordered integral domains
	4.3. The field of real numbers
	4.4. The field of complex numbers
5. Groups Again
	5.1. Permutations
	5.2. Cyclic groups
	5.3. Direct products of groups
	5.4. Finite abelian groups
6. Polynomial Rings
	6.1. Polynomial rings
	6.2. Roots, divisibility, and the greatest common divisor
	6.3. Polynomials in Q[x], R[x], and C[x]
7. Modular Arithmetic in F[x] and Unique Factorization Domains
	7.1. Modular arithmetic in F[x]
	7.2. Euclidean domains, principal ideal domains, unique factorization domains
	7.3. Normed domains
8. Field Extensions
	8.1. Bases of field extensions
	8.2. Simple field extensions
	8.3. Algebraic field extensions
	8.4. Splitting fields; algebraically closed fields
	8.5. Geometric constructions and the famous problems of antiquity
9. An Introduction to Galois Theory
	9.1. Classical formulas
	9.2. Solvable groups
	9.3. Solvability by radicals
	9.4. The fundamental theorem of Galois theory
10. Vector Spaces
	10.1. Vector spaces
	10.2. Linear transformations
	10.3. Inner product spaces
11. Modules
	11.1. Modules
	11.2. Modules of quotients
	11.3. Free modules
Answers and Hints to Selected Problems
Index
Errata