A Book of Abstract Algebra, Second Edition by Charles C. Pinter is a comprehensive guide to abstract algebra and its applications. The book provides a clear and in-depth explanation of the concepts and techniques involved in abstract algebra, making it an ideal resource for students and professionals alike.
The second edition of this popular book has been updated to include recent developments in the field, making it even more relevant and useful to its readers. The book covers a wide range of topics, from the basics of abstract algebra to more advanced concepts, including groups, rings, fields, and Galois theory. Each chapter is clearly written and contains numerous examples and exercises to help reinforce the material.
One of the key strengths of A Book of Abstract Algebra is its focus on the underlying concepts and ideas. The author presents the material in a way that is both accessible and rigorous, making it ideal for students with a background in mathematics who are looking to expand their knowledge of abstract algebra. The book also provides a solid foundation for students who wish to pursue further study in related fields, such as number theory or algebraic geometry.
TABLE OF CONTENTS
Chapter 1 Why Abstract Algebra?
- History of Algebra
- New Algebras
- Algebraic Structures
- Axioms and Axiomatic Algebra
- Abstraction in Algebra
Chapter 2 Operations
- Operations on a Set
- Properties of Operations
Chapter 3 Groups
- Examples of Infinite and Finite Groups
- Examples of Abelian and Non-abelian Groups
- Group Tables
Chapter 4 Elementary Properties of Groups
- Uniquness of Identity and Inverses
- Properties of Inverses
- Direct Product of Groups
Chapter 5 Subgroups
- Definition of Subgroup
- Generators and Defining Relations
- Cay Ley Diagrams
- Centre of a Group
- Group Codes
- Hamming Code
.
Chapter 6 Functions
- Injective
- Surjective
- Bijective Function
- Composite and Inverse of Functions
- Finite State Machines
- Automata and their Semigroups
Chapter 7 Groups of Permutations
- Symmetric Groups
- Dihedral Groups
- An Application of Groups to Anthropology
Chapter 8 Permutations of a Finite Set
- Decomposition of Permutations into Cycles
- Transpositions
- Even and Odd Permutations
- Alternating Groups
Chapter 9 Isomorphism
- The Concept of Isomorphism in Mathematics
- Isomorphic and Non-Isomorphic Groups
- Cayley’s Theorem
Chapter 10 Order of Group Elements
- Powers/Multiples of Group Elements
- Laws of Exponents
- Properties of the order of Group Elements
Chapter 11 Cyclic Groups
- Finite and Infinite Cyclic Groups
- Isomorphism of Cyclic Groups
- Subgroups of Cyclic Groups
Chapter 12 Partitions and Equivalence Relations
Chapter 13 Counting Cosets
- Lagrange’s Theorem and Elementry Consequences
- Survey of Groups of Order ≤10
- Number of Conjugate Elements
- Group Acting on a Set
Chapter 14 Homomorphisms
- Elementry Properties of Homomorphisms
- Normal Subgroups
- Kernel and Range
- The Class Equation
- Induction on the Order of a Group
Chapter 16 The Fundamental Homomorphism Theorem
- Fundamental Homomorphism Theorem and some Consequences
- The Isomorphism Theorems
- The Correspondence Theorem
- Cauchy’s Theorem
- Sylow Subgroups
- Sylow’s Theorem
- Decomposition Theorem for Finite Abelian Groups
Chapter 17 Rings: Definitions and Elementry Properties
- Commutative Rings
- Unity
- Invertibles and Zero Divisors
- Integral Domain Field
Chapter 18 Ideals and Homomorphisms
Chapter 19 Quotient Rings
- Construction of Quotient Rings, Examples
- Fundamental Homomorphism Theorem and some Consequences
- Properties of Prime and Maximal Ideals
Chapter 20 Integral Domains
- Characteristics of an Integral Domain
- Properties of the Characteristics
- Finite Fields
- Construction of the Fields of Quotients
Chapter 21 The Integers
- Ordered Integral Domains
- Well-Ordering
- Characterization of z Up to Isomorphism
- Mathematical Induction
- Division Algorithm
Chapter 22 Factoring into Primes
- Ideals of z
- Properties of the GCD
- Relatively Prime Integers
- Primes
- Euclid’s Lemma
- Unique Factorization
Chapter 23 Elements of Number Theory
- Properties of Congruence
- Theorems of Fermat and Euler
- Solutions of Linear Congruences
- Chinese Remainder Theorem
- Wilson’s Theorem and Consequences
- Quadratic Residues
- The Legendre Symbol
- Primitive Roots
Chapter 24 Rings of Polynomials
- Motivation and Definition
- Domain of Polynomials over a Field
- Division Algorithm
- Polynomials in Several Variables
- Fields of Polynomial Quotients
Chapter 25 Factoring Polynomials
- Ideals of F[x]
- Properties of GCD
- Irreducible Polynomials
- Unique Factorization
- Euclidean Algorithm
Chapter 26 Substitution in Polynomials
- Roots and Factors
- Polynomial Functions
- Polynomial over Q
- Eisenstein’s Irreducibility Criterion
- Polynomials over the Reals
- Polynomial Interpolation
Chapter 27 Extensions of Fields
- Algebraic and Transcendental Elements
- The Minimum Polynomial
- Basic Theorem on Field Extensions
Chapter 28 Vector Spaces
- Elementary Properties of Vector Spaces
- Linear Independence
- Basis
- Dimension
- Linear Transformations
Chapter 29 Degrees of Field Extensions
- Simple and Iterated Extensions
- Degree of an Iterated Extensions
- Fields of Algebraic Elements
- Algebraic Numbers
- Algebraic Closure
Chapter 30 Ruler and Compass
- Constructible Points and Numbers
- Impossible Constructions
- Constructible Angles and Polygons
Chapter 31 Galois Theory: Preamble
- Multiple Roots
- Root Field
- Extension of a Field
- Isomorphism
- Roots of Unity
- Separable Polynomials
- Normal Extensions
Chapter 32 Galois Theory: The Heart of the Matter
- Field Automorphisms
- The Galois Group
- The Galois Correspondence
- Fundamental Computing Galois Groups
Chapter 33 Solving Equations by Radicals
- Radical Extensions
- Abelian Extensions
- Solvable Groups
- Insolvability of the Quin tic
Download Now Book in PDF
In addition to its clear and comprehensive coverage of abstract algebra, the book also includes a wealth of supplementary material that is designed to help students better understand and retain the material. This includes appendixes on set theory, relations and functions, and Boolean algebra, as well as numerous examples and exercises that are designed to reinforce the concepts presented in each chapter.
For students looking to access the book’s material electronically, A Book of Abstract Algebra, Second Edition is available in PDF format for download. This makes it easy for students to access the book from anywhere, at any time, and on a variety of devices. The PDF format also provides the added convenience of being searchable and easily navigable, making it easier for students to find the information they need quickly and efficiently.
In conclusion, A Book of Abstract Algebra, Second Edition by Charles C. Pinter is an excellent resource for anyone looking to learn about abstract algebra. Whether you are a student just starting out in the field or a professional looking to brush up on your knowledge, this book provides a clear, comprehensive, and accessible guide to the subject. With its focus on the underlying concepts and its wealth of supplementary material, this book is sure to be a valuable resource for years to come.
