A First Course in Abstract Algebra: With Applications

Title Page
Special Notation
Contents
Preface
1 Number theory
	1.1 Induction
	1.2 Binomial Coefficients
	1.3 Greatest Common Divisors
	1.4 The Fundamental Theorem of Arithmetic
	1.5 Congruences
	1.6 Dates and Days
2 Groups I
	2.1 Some Set Theory
		Functions
		Equivalence relations
	2.2 Permutations
	2.3 Groups
		Symmetry
	2.4 Subgroups and Lagrange\'s Theorem
	2.5 Homomorphisms
	2.6 Quotient Groups
	2.7 Group Actions
	2.8 Counting with Groups
3 Rings I
	3.1 First properties
	3.2 Fields
	3.3 Polynomials
	3.4 Homomorphisms
	3.5 Greatest Common Divisors
		Euclidean Rings
	3.6 Unique Factorization
	3.7 Irreducibility
	3.8 Quotient Rings and Finite Fields
	3.9 Officers, Magic, Fertilizer, and Horizons
		Officers
		Magic
		Fertilizer
		Horizons
4 Linear Algebra
	4.1 Vector Spaces
		Gaussian Elimination
	4.2 Euclidean Constructions
	4.3 Linear Transformations
	3.4 Determinants
	3.5 Codes
		Block Codes
		Linear Codes
5 Fields
	4.1 Classical Formulas
		Viète\'s Cubic Formula
	4.2 Insolvability of the General Quintic
		Formulas and Solvability by Radicals
		Translation into Group Theory
	5.3 Epilog
6 Groups II
	6.1 Finite Abelian Groups
	6.2 The Sylow Theorems
	6.3 Ornamental Symmetry
7 Commutative Rings II
	7.1 Prime Ideals and Maximal Ideals
	7.2 Unique Factorization
	7.3 Noetherian Rings
	7.4 Varieties
	7.5 Gröbner Bases
		Monomial Orders
		Generalized Division Algorithm
		Gröbner Bases
Appendix A: Inequalities
Appendix B: Pseudocodes
Hints for Selected Exercises
	Chapter 1
	Chapter 2
	Chapter 3
	Chapter 4
	Chapter 5
	Chapter 6
	Chapter 7
Bibliography
Index