A Course In Algebra (Paperback) by Vijay K Khanna, S K Bhambri

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Book Summary of A Course In Algebra

Designed for undergraduate and post graduate students of mathematics, the book can also be used by those preparing for various competitive examinations. The text starts with a brief introduction to results from Set Theory and Number Theory. It then goes on to cover Groups, Rings, Vector spaces and Fields. The topics under Groups include Subgroups, Normal subgroups, Finitely generated abelian groups, Group actions, Solvable and Nilpotent groups. The course in Ring Theory covers Ideas, Imbedding of rings, Euclidean domains, Principal ideal domains, Unique factorization domains, Polynomial rings, Noetherian (Artinian) rings. The section on Vector spaces deals with Linear transformations, Inner product spaces, Dual spaces, Eigen spaces, Diagonalizable operators etc. Under Fields, Algebraic extensions, Splitting fields, Normal extensions, Separable extensions, Algebraically closed fields, Galois extensions and construction by ruler and compass are discussed. The theory has been strongly supported by numerous examples and worked out problems. There is plenty of scope also for the reader to try and solve problems on his (her) own.

Salient Features
New In the Third Edition:

• A full new chapter on Groups
• Revamping of the chapter on Eigen Values to make it more reader-friendly
• Splitting of the chapter on Fields for a focused approach.

About The Author
Vijay K Khanna is Reader in Deptt. of Mathematics, Kirori Mal College, University of Delhi and has been teaching undergraduate and postgraduate students for over 35 years. His other publications include Lattices and Boolean Algebras, Solid Geometry, and Business Mathematics published by Vikas.

S K Bhambri is Reader in Deptt. of Mathematics, Kirori Mal College, University of Delhi and has been teaching undergraduate and postgraduate students for over 35 years. He got his doctorate in 1981 from London. He is co-author of Business Mathematics, published by Vikas.

Table Of Contents
Preface to the Third Edition
Preface to the First Edition
Glossary of Symbols
1. Preliminaries

• Sets
• Subsets
• Relations
• Equivalence Classes
• Mappings or Functions
• Equality of Mappings
• Composition of Mappings
• Binary Compositions
• Permutations
• Cyclic Permutations
• Cycles of a Permutation
• Disjoint Permutations
• Some Results From Number Theory
• The Greatest Common Divisor
• Prime Numbers
• Composite Numbers
• Congruences

2. Groups

• Semi groups
• Subgroups
• Cyclic Groups

3. Normal Subgroups, Homomorphisms, Permutation Groups

• Quotient Groups
• Homomorphisms ? Isomorphisms
• The Dihedral Group
• Permutation Groups
• Generators of a Subgroup
• Cummutator

4. Automorphisms and Conjugate Elements

• Inner Automorphisms
• Characteristic Subgroups
• Conjugate Elements
• Similar Permutations
• Partition of Integer

5. Sylow Theorems and Direct Products

• Sylow p-subgroups
• Double Cosets
• Sylow Groups in Sp'
• Direct Products
• Finite Abelian Groups

6. Group Actions, Solvable and Nilpotent Groups

• Group Actions
• Normal Series
• Solvable Groups
• Nilpotent Groups

7. Rings

• Subrings
• Sum of Two Subrings
• Characteristic of a Ring
• Product of Rings
• Ideals
• Sum of Two Ideals
• Product of Two Ideals

8. Homomorphisms and Imbedding of Rings

• Quotient Rings
• Homomorphisms
• Imbedding of Rings
• More on Ideals
• Maximal Ideals

9. Euclidean and Factorization Domains

• Euclidean Domains
• Prime and Irreducible Elements
• Polynomial Rings
• Greatest Common Divisor
• Unique Factorization Domains
• Noetherian Rings

10. Vector Spaces

• Subspaces
• Sum of Subspaces
• Quotient Spaces
• Homomorphisms of Linear Transformation
• Linear Span
• Linear Dependence and Independence
• Inner Product Spaces
• Norm of Vector
• Orthogonality
• Orthonormal Set

11. Linear Transformations

• Algebra of Linear Transformations
• Invertible Linear Transformations
• Matrix of a Linear Transformation
• Dual Spaces
• Transpose of a Linear Transformation

12. Eigen Values and Eigen Vectors

• Characteristic Polynomials
• Characteristic Polynomial of a Linear Operator
• Minimal Polynomials
• Diagonalizable Operators
• Primary Decomposition Theorem
• Invariant Subspaces
• Projections

13. Fields

• Algebraic Extensions
• Roots of Polynomials
• Splitting Fields
• Ruler and Compass Constructions

14. More on Fields

• Prime Sub fields
• Separable Extensions
• Normal Extensions
• Algebraically Closed Fields and Algebraic Closure
• Automorphisms of Field Extensions
• Galois Extensions
• Roots of Unity
• Finite Fields

Index

Details of Book: A Course In Algebra

Book:	A Course In Algebra
Author:	Vijay K Khanna, S K Bhambri
ISBN:	8125919112
ISBN-13:	9788125919117 ,  978-8125919117
Binding:	Paperback
Publishing Date:	2008
Publisher:	Vikas Publishing House
Edition:	3rdEdition
Number of Pages:	676
Language:	English