Advanced Abstract Algebra

Paper Code: 
MAT 121
Credits: 
5
Contact Hours: 
60.00
Objective: 
  • To introduce basics in mathematics.
  • To introduce and develop abstract concepts.
Unit I: 

Direct product of groups (External and Internal). Isomorphism theorems - Diamond isomorphism theorem, Butterfly Lemma, Conjugate classes, Sylows theorm, p- sylow theorem .

Unit II: 

Commutators, Derived subgroups, Normal series and Solvable groups, Composition series, Refinement theorem and Jordan-Holder theorem for infinite groups.

Unit III: 

Polynomial rings, Euclidean rings. Modules, Submodules, Quotient modules Direct sums and Module Homomorphisms. Generation of modules, Cyclic modules.

Unit IV: 

Field theory - Extension fields, Algebraic and Transcendental extensions, Separable and inseparable extensions, Normal extensions. Splitting fields. Galois theory - the elements of Galois theory, Fundamental theorem of Galois theory,

Unit V: 

Galois theory - the elements of Galois theory, Fundamental theorem of Galois theory , Solvalibility by radicals.

Essential Readings: 
  1. Chauhan & Singh, Advanced Abstract Algebra, JPH, Jaipur.
  2. P.B.Bhattacharya, S.K.Jain and S.R.Nagpaul,Basic Abstract Algebra , Cambridge University Press.
  3. I.N.Herstein , Topics in Algebra , Wiley Eastern Ltd., New Delhi
References: 
  1. Deepak Chatterjee,Abstract Algebra.PHI. Ltd.New Delhi.
  2. Malcolm Birkoff ,Abstract Algebra , Cambridge University Press