Krishna's - Abstract Algebra 3.2

SYLLABUS- ABSTRACT ALGEBRA, Unit-I
Automorphism, inner automorphism, automorphism groups and their
computations. Conjugacy relations, Normaliser, Counting principle and the
class equation of a finite group, Center of group of prime power order, Sylow's
theorems, Sylow p-subgroup.
Unit-II
Prime and maximal ideals, Euclidean Rings, Principal ideal rings, Polynomial
Rings, Polynomial over the Rational Field, The Eisenstein Criterion, Polynomial
Rings over Commutative Rings, unique factorization domain, R is unique
factorization domain implies so is R [x1, x2, ..., x n].
Unit-III
Direct sum, Quotient space, Linear transformations and their representation as
matrices, The Algebra of linear transformations, rank nullity theorem, change of
basis, linear functional, Dual space, Bidual space and natural isomorphism,
transpose of a linear transformation, Characteristic values, annihilating
polynomials, diagonalisation, Cayley Hamilton Theorem, Invariant subspaces,
Primary decomposition theorem.
Unit-IV
Inner product spaces, Cauchy-Schwarz inequality, orthogonal vectors,
Orthogonal complements, Orthonormal sets and bases, Bessel's inequality for
finite dimensional spaces, Gram-Schmidt orthogonalization process, Bilinear,
Quadratic and Hermitian forms.