Abstract Algebra, 2nd Edition, written by Lloyd R. Jaisingh and Frank Ayres, is a comprehensive textbook designed to help students understand the advanced concepts of abstract algebra. This book covers a wide range of topics, including group theory, ring theory, field theory, and Galois theory.
One of the key features of the Abstract Algebra, 2nd Edition, book is its clear and concise writing style. The authors have done an excellent job of breaking down complex mathematical concepts into easy-to-understand terms and explanations. Additionally, the book is filled with a variety of examples, illustrations, and practice problems that help students reinforce what they have learned and gain a deeper understanding of the material.
Another important aspect of the Abstract Algebra, 2nd Edition, book is its comprehensive approach to teaching abstract algebra. The book covers all of the topics that students will need to know in order to succeed in advanced mathematics and beyond. Whether a student is just starting to learn abstract algebra or has been studying it for a while, the Abstract Algebra, 2nd Edition, book has something to offer.
TABLE OF CONTENTS
Unit 1 Sets
- Sets
- Equal Sets
- Subsets of a Set
- Universal Sets
- Intersection and Union of Sets
- Venn Diagrams
- Operations with Sets
- The Product Sets
- Mappings
- One-to-One Mappings
- One-to-One Mapping of a Set onto itself
Chapter 2 Relations and Operations
- Relations
- Properties of Binary Relations
- Equivalence Sets
- Ordering in Sets
- Operations
- Types of Binary Operations
- Well-defined Operations
- Isomorphisms
- Permutations
- Transpositions
- Algebraic Systems
Chapter 3 The Natural Numbers
- The Peano Postulates
- Addition on N
- Multiplication on N
- Mathematical Induction
- The Order Relations
- Multiples and Powers
- Isomorphic Sets
.
Chapter 4 The Integers
- Binary Relations
- Addition and Multiplication on J
- The positive Integers
- Zero and Negative Integers
- The Integers
- Order Relations
- Subtraction
- Absolute Value |a|
- Addition and Multiplication on Z
- Other Properties of Integers
Chapter 5 Some Properties of Integers
- Divisors
- Primes
- Greatest Common Divisor
- Relatively Prime Integers
- Prime Factors
- Congruence’s
- The Algebra of Residue Classes
- Linear Congruence’s
- Positional Notation for Integers
Chapter 6 Rational Numbers
- The Rational Numbers
- Addition and Multiplication
- Subtraction and Division
- Replacement
- Order Relations
- Reduction to Lowest Terms
- Decimal Representation
Chapter 7 The Real Numbers
- Dedekind Cuts
- Positive Cuts
- Multiplicative Inverses
- Additive Inverses
- Multiplication on K
- Subtraction and Division
- Order Relations
- Properties of the Real Numbers
Chapter 8 The Complex Numbers
- Addition and Multiplication on C
- Properties of Complex Numbers
- Subtraction and Division of C
- Trigonometric Representation
- Roots
- Primitive Roots of Unity
Chapter 9 Groups
- Groups
- Simple Properties of Groups
- Subgroups
- Cyclic Groups
- Permutation Groups
- Homomorpisms
- Isomorphisms
- Cosets
- Invariant Subgroups
- Quotient Groups
- Product of Subgroups
- Composition Series
Chapter 10 Further Topics on Group Theory
- Cauchy’s Theorem for Groups
- Groups of Order 2p
- The Sylow Theorems
- Galois Group
Chapter 11 Rings
- Rings
- Properties of Rings
- Subrings
- Types of Rings
- Characteristic
- Divisors of Zero
- Homomorphisms and Isomorphisms
- Ideals
- Principal Ideals
- Prime and Maximal Ideals
- Quotient Rings
- Euclidean Rings
Chapter 12 Integral Domains, Division Rings, Fields
- Integral Domains
- Unit, Associate, Divisor
- Subdomains
- Ordered Integral Domains
- Division Algorithm
- Unique Factorization
- Division Rings
- Fields
Chapter 13 Polynomials
- Polynomial Forms
- Monic Polynomials
- Division
- Commutative Polynomial Rings with Unity
- Substitution Process
- The Polynomial Domain F[x]
- Prime Polynomials
- The Polynomial Domain C[x]
- Greatest Common Divisor
- Properties of the Polynomial Domain F[x]
Chapter 14 Vector Spaces
- Vector Spaces
- Subspaces of a Vector Space
- Linear Dependence
- Bases of a Vector Space
- Subspaces of a Vector Space
- Vector Spaces Over R
- Linear Transformations
- The Algebra of Linear Transformations
Chapter 15 Matrices
- Matrices
- Square Matrices
- Total Matrix Algebra
- Amatrix of Order mxn
- Solutions of a System of Linear Equations
- Elementary Transformations on a Matrix
- Upper Triangular, Lower Triangular, and Diagonal Matrices
- A Canonical Form
- Elementary Column Transformations
- Elementary Matrices
- Inverses of Elementary Matrices
- The Inverse of a Non-Singular Matrix
- Minimum Polynomial of a Square Matrix
- Systems of Linear Equations
- Systems of Non-Homogeneous Linear Equations
- Systems of Homogeneous Linear Equations
- Determinant of a Square Matrix
- Properties of Determinants
- Evaluation of Determinants
Chapter 16 Matrix Polynomials
- Matrices with Polynomials Elements
- Elementary Transformations
- Polynomials with Matrix Coefficients
- Division Algorithm
- The Characteristic Roots and Vectors of a Matrix
- Similar Matrices
- Real Symmetric Matrices
- Orthogonal Matrices
- Conics and Quadratic Surfaces
Chapter 17 Linear Algebra
- An Isomorphsim
- Chapter 18 Boolean Algebras
- Boolean algebra
- Boolean Functions
- Normal Forms
- Changing the form of a Boolean Function
- Order Relation in a Boolean Algebra
- Algebra of Electrical Networks
- Simplification of Networks
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In conclusion, Abstract Algebra, 2nd Edition, by Lloyd R. Jaisingh and Frank Ayres is an excellent resource for students who are looking to master the advanced concepts of abstract algebra. With its clear and concise writing style, comprehensive approach to teaching, and user-friendly design, the Abstract Algebra, 2nd Edition, book is the perfect tool for students who want to excel in mathematics. So, if you are looking for a reliable and effective guide to help you master abstract algebra, be sure to check out the Abstract Algebra, 2nd Edition, book today!
