Modern algebra or you can say abstract algebra pdf by krishna series is especially designed for Degree, Honours and Postgraduate Students of all Indian Universities, which is also helpful in various undergraduate and post-graduate level competition exams such as IITJAM, CSIR-NET, UPSC etc for math paper.

The book is written by A R Vasishta who is a very popular author for Higher mathematics books such as linear algebra, matrices and many more. A R vasishtha is from department of mathematics in Meerut college, Meerut.

## Chapter content of modern algebra by a r vasishtha pdf:

01. Some Basic Set Theoretic  Concepts

• Mathematical logic
•        Tautologies
•        Set
•        Subsots of a set
•        Union of sets
•        Intersection of sets
•        Cartesian product of two sets
•        Functions or mappings
•        Binary operation
•        Relations
•        Equivalence relations
•        Equivalence classes
•        Partitions
•        Partial order relations

02. Groups

•         Binary operation
•        Algebraic structure
•        Group. Definition
•        Abolian group
•        Finite and infinite groups
•        Order of a finite group
•        General properties of groups
•        Definition of a group based upon left axioms
•        Composition tables for finite sets
•        Multiplication modulo p
•        Residue classes of the set of integors
•        An alternative set of postulates for a group
•        Permutations
•        Groups of permutations
•        Cyclic per mutations
•        Even and odd permutations
•        Integral powers of an element of a group
•        Order of an element of a group
•        Isomorphism of groups
•        The relation of isomorphism in the set of all groups
•        Complexes and subgroups of a group
•        Intersection of subgroups
•        Co-sets
•        Relation of congruence modulo
•        Lagrange’s theorem
•        Euler’s theorem
•        Fermat’s theorem
•        Order of the product of  2 subgroups of finite order
•        Cayley’s theorem
•        Cyclic groups
•        Subgroup generated by a subset of a group
•        Generating systens of a group

03. Groups (Continued)

•       Normal subgroups
•       Conjugate elements
•       Normalizer of an element of a group
•       Class equation of a group
•       Centre of a group
•       Conjugate subgroups
•       Invariant subgroups
•       Quotient groups
•       Homomorphism of groups
•       Kernel of a homomorphism
•       Fundamental theorem on homomorphism of groups
•       Automorphisms of a group
•       Inner automorphisms
•       More results on group homomorphism
•       Maximal supergroups
•       Composition series of a group and the Jordan-Holder theorem
•       Solvable groups
•       Commutator subgroup of a group
•       Direct products
•       External direct products
•       Internal direct products
•       Cauchy’s theorem on abelian groups
•       Cauchy’s theorem
•       Sylow’s theorem

04. Rings

•       Ring. Definition
•       Elementary properties of a ring
•       Rings with or without 0 divisors
•       Integral domain
•       Field
•       Division ring or skew field
•       Isomorphism of rings
•       Subrings
•       Subfields
•       Characteristic of a ring
•       Ordered integral domains
•       Imbedding of a ring into another ring
•       The field of quotients
•       Ideals
•       Principal ideal
•       Principlal ideal ring
•       Divisibility in an integral domain
•       Units
•       Associates
•       Prime elements
•       Greatest common divisor
•       Polynomial rings
•       Polynomials over an integral domain
•       Division algorithm for polynomials over a field
•       Euclidean algorithm for polynomials over a field
•       Unique factorization domain
•       Unique factorization theorem for polynomials over a field
•       Remainder theorem
•       Prime fields
•       Rings of endomorphisms of an abelian group

05. Rings (Continued)

•       Quotient rings or residue class rings
•       Homomorphism of rings
•       Kernel of a ring homomorphism
•       Maximal ideals
•       Prime ideals
•       Euclidean rings or Euclidean domains
•       Polynomial rings over unique factorization domains

06. Vector Space

•       Vector space Definition
•       General properties of vector spaces
•       Vector subspaces
•       Linear combination of vectors
•       Linear span
•       Linear sum of two subspaces
•       Linear dependence and linear independence of vectors
•       Basis of a vector space
•       Finite dimensional vector spaces
•       Dimension of a finitely generated vector space
•       Homomorphism of vector spaces or Linear tradsformation
•       Isomorphism of vector spaces
•       Quotient space
•       Direct sum of spaces
•       Complementary subspaces
•       Co-ordinates

07. Vector Space (Continued)

•        Linear transformations as vectors
•        Dual space
•        Dual basis
•        Reflexivity
•        Annihilators

08. Modules

•        Modules. Definition
•        Submodules
•        Direct sum of submodules
•        Homomorphism of modules or linear transformations
•        Quotient modules
•        Cyclic modules
•        Fundamental theorem on finitely generated  modules over Euclidean rings

09. Extension Fields and Galois Theory

•       Field extensions
•       Finite field extension
•       Simple field extension
•       Algebraic field extensions
•       Transcendental element
•       Roots of polynomials
•       Multiple roots
•       Splitting field or decomposition field
•       Uniqueness of tho splitting field
•       Derivative of a polynomial
•       Separable extension
•       Perfect field
•       The elements of Galois theory
•       Fixed field
•       Normal extension
•       Galois group
•       Fundamental theorem of Galois theory
•       Construction with ruler and compass
•       Finite fields

Index