Abstract Algebra is a fundamental branch of mathematics that is studied at the undergraduate and graduate levels. Understanding the concepts of abstract algebra is crucial for students who wish to pursue a career in mathematics, computer science, or any related field. The book “Basic Abstract Algebra” 2nd Edition by PB Bhattacharya, S.K Jain, and S.R Naugpal is an essential resource for students who wish to master the subject.
PB Bhattacharya, S.K Jain, and S.R Naugpal are well-respected mathematicians and researchers with extensive experience in the field of abstract algebra. They have collaborated to produce a comprehensive guide to the subject that is both accessible and informative.
The book “Basic Abstract Algebra” covers all the essential topics of abstract algebra, starting from the basics and progressing to more advanced topics. The authors have used clear and concise language to explain complex mathematical concepts, making the book accessible to students of all levels. The book also includes numerous examples and exercises to help students understand and apply the concepts they learn.
TABLE OF CONTENTS
Chapter I Sets and mappings
- Sets
- Relations
- Mappings
- Binary operations
- The cardinality of a set
Chapter 2 Integers, real numbers, and complex numbers
- Integers
- Rational, real, and complex numbers
- Fields
Chapter 3 Matrices and determinants
- Matrices
- Operations on matrices
- Partitions of a matrix
- The determinant function
- Properties of the determinant function
- Expansion of det A
Chapter 4 Groups
- Semigroups and groups
- Homomorphisms
- Subgroups and cosets
- Cyclic groups
- Permutation groups
- Generators and relations
Chapter 5 Normal subgroups
- Normal subgroups and quotient groups
- Isomorphism theorems
- Automorphisms
- Conjugacy and G-sets
Chapter 6 Normal series
- Normal series
- Solvable groups
- Nilpotent groups
.
Chapter 7 Permutation groups
- Cyclic decomposition
- Alternating group
Chapter 8 Structure theorems of groups
- Direct products
- Finitely generated abelian groups
- Invariants of a finite abelian group
- Sylow theorems
- Groups of orders p2
Chapter 9 Rings
- Definition and examples
- Elementary properties of rings
- Types of rings
- Subrings and characteristic of a ring
- Additional examples of rings
Chapter 10 Ideals and homomorphisms
- Ideals
- Homomorphisms
- Sum and direct sum of ideals
- Maximal and prime ideals
- Nilpotent and nil ideals
- Zorn’s lemma
Chapter 11 Unique factorization domains and Euclidean domains
- Unique factorization domains
- Principal ideal domains
- Euclidean domains
- Polynomial rings over UFD
Chapter 12 Rings of fractions
- Rings of fractions
- Rings with Ore condition
Chapter 13 Integers
- Peano’s axioms
- Integers
Chapter 14 Modules and vector spaces
- Definition and examples
- Submodules and direct sums
- R-homomorphisms and quotient modules
- Completely reducible modules
- Free modules
- Representation of linear mappings
- Rank of a linear mapping
Chapter 15 Algebraic extensions of fields
- Irreducible polynomials and Eisenstein criterion
- Adjunction of roots
- Algebraic extensions
- Algebraically closed fields
Chapter 16 Normal and separable extensions
- Splitting fields
- Normal extensions
- Multiple roots
- Finite fields
- Separable extensions
Chapter 17 Galois theory
- Automorphism groups and fixed fields
- Fundamental theorem of Galois theory
- Fundamental theorem of algebra
Chapter 18 Applications of Galois theory to classical problems
- Roots of unity and cyclotomic polynomials
- Cyclic extensions
- Polynomials solvable by radicals
- Symmetric functions
- Ruler and compass constructions
Chapter 19 Noetherian and Artinian modules and rings
- HomR
- Noetherian and Artinian modules
- Wedderburn—Artin theorem
- Uniform modules, primary modules, and Noether—Lasker theorem
Chapter 20 Smith normal form over a PID and rank
- Preliminaries
- Row module, column module, and rank
- Smith normal form
Chapter 21 Finitely generated modules over a PID
- Decomposition theorem
- Uniqueness of the decomposition
- Application to finitely generated abelian groups
- Rational canonical form
- Generalized Jordan form over any field
Chapter 22 Tensor products
- Categories and functors
- Tensor products
- Module structure of tensor product
- Tensor product of homomorphisms
- Tensor product of algebras
Download Now Book in PDF
One of the key features of the book is its comprehensive coverage of the subject. The authors have included a wide range of topics, starting from the fundamentals of groups, rings, and fields and progressing to more advanced topics such as Galois theory and modules. The book is well-structured, making it easy for students to follow and understand the concepts.
In conclusion, the book “Basic Abstract Algebra” 2nd Edition by PB Bhattacharya, S.K Jain, and S.R Naugpal is an essential resource for students who wish to master the subject of abstract algebra. The authors have done an excellent job of covering all the essential topics in a clear and concise manner, making the book accessible to students of all levels. We are pleased to inform students that this book is available in PDF format, making it easy for students to download and access.
