Fundamentals of Abstract Algebra

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
SECTION I: Groups
	CHAPTER 1: Background Material
		1.1. EQUIVALENCE RELATIONS
		1.2. FUNCTIONS
		1.3. BASIC NUMBER THEORY
		1.4. MODULO ARITHMETIC
	CHAPTER 2: Basic Group Theory
		2.1. DEFINITIONS AND EXAMPLES
			2.1.1. Groups of Small Order
			2.1.2. Group Exponentiation
		2.2. SUBGROUPS
		2.3. CYCLIC GROUPS
		2.4. PERMUTATION GROUPS
		2.5. PRODUCTS OF GROUPS
		2.6. HOMOMORPHISMS
		2.7. ISOMORPHIC GROUPS
		2.8. COSETS OF A GROUP
		2.9. FACTOR GROUPS AND NORMAL SUBGROUPS
			2.9.1. Semidirect Products
		2.10. NORMAL AND SIMPLE GROUPS
		2.11. THE GROUP ISOMORPHISM THEOREMS
	CHAPTER 3: Simple Groups
		3.1. THE ALTERNATING GROUP
		3.2. THE PROJECTIVE LINEAR GROUPS
	CHAPTER 4: Group Action
		4.1. GROUP ACTION ON A SET
		4.2. BURNSIDE’S LEMMA
		4.3. POLYA’S FORMULA
		4.4. SOME CONSEQUENCES OF GROUP ACTION
		4.5. SYLOW THEORY
		4.6. CLASSIFYING FINITE GROUPS WITH SYLOW THEORY
		4.7. FINITE ABELIAN GROUPS
	CHAPTER 5: Group Presentation and Representations
		5.1. FREE GROUPS
		5.2. GROUP PRESENTATIONS
		5.3. GROUP REPRESENTATION
	CHAPTER 6: Solvable and Nilpotent Groups
		6.1. SOME RELEVANT SUBGROUPS
		6.2. SERIES OF GROUPS
		6.3. SOLVABLE AND NILPOTENT GROUPS
SECTION II: Rings and Fields
	CHAPTER 7: Ring Theory
		7.1. DEFINITION AND EXAMPLES
		7.2. INTEGRAL DOMAINS
		7.3. THE QUATERNIONS
		7.4. RING HOMOMORPHISMS
		7.5. FACTOR RINGS AND IDEALS
		7.6. QUOTIENT FIELD OF AN INTEGRAL DOMAIN
		7.7. CHARACTERISTIC OF A RING
		7.8. THE RING OF POLYNOMIALS
		7.9. SPECIAL IDEALS
	CHAPTER 8: Integral Domain Theory
		8.1. EUCLIDEAN AND PRINCIPAL IDEAL DOMAINS
		8.2. UNIQUE FACTORIZATION DOMAINS
		8.3. ONE PARTICULAR INTEGRAL DOMAIN
		8.4. POLYNOMIALS OVER A UFD
	CHAPTER 9: Field Theory
		9.1. REVIEW AND ALGEBRAICITY
		9.2. VECTOR SPACES & EXTENSION FIELDS
		9.3. GEOMETRIC CONSTRUCTIONS
			9.3.1. Famous Impossibilities
		9.4. ALGEBRAIC EXTENSION & CLOSURE
		9.5. EXISTENCE THEOREMS
		9.6. FINITE FIELDS
	CHAPTER 10: Galois Theory
		10.1. FIELD HOMOMORPHISMS
		10.2. COMPUTING GALOIS GROUPS
		10.3. APPLICATIONS OF ZORN’S LEMMA
		10.4. TWO IMPORTANT THEOREMS
		10.5. SEPARABLE DEGREE
		10.6. GALOIS EXTENSIONS
		10.7. SOME PRELIMINARY THEOREMS
		10.8. THE FUNDAMENTAL THEOREM OF GALOIS THEORY
		10.9. SOLVABLE GROUP ESSENTIALS
		10.10. SOLVABILITY BY RADICALS
References
Index