Fundamentals of Abstract Algebra is a primary textbook for a one year first course in Abstract Algebra, but it has much more to offer besides this. The book is full of opportunities for further, deeper reading, including explorations of interesting applications and more advanced topics, such as Galois theory. Replete with exercises and examples, the book is geared towards careful pedagogy and accessibility, and requires only minimal prerequisites. The book includes a primer on some basic mathematical concepts that will be useful for readers to understand, and in this sense the book is self-contained. Features • Self-contained treatments of all topics. • Everything required for a one-year first course in Abstract Algebra, and could also be used as supplementary reading for a second course. • Copious exercises and examples.
Author(s): Mark J. DeBonis
Publisher: Chapman and Hall/CRC
Year: 2024
Language: English
Pages: 302
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
