ADVANCED ALGEBRA

Paper Code: 
MAT121
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 
This course will enable the students to -
  1. Demonstrate knowledge of conjugacy relation and class equation. 
  2. Apply Sylow theorem to determine the nature of subgroups. 
  3. Identify the irreducibility of polynomials.
  4. Develop the concepts of extension fields. 
  5. Find the splitting field for a given polynomial.

   Learning Outcomes

Learning and teaching strategies

Assessment

After the completion of the course the students will be able to:

CLO1- understand and introduce the language and precision of abstract algebra.

CLO2- The course is proof-based, in the sense that students will be expected to understand, construct, and write proofs.

CLO3-. The course will create the tendency to think of why a mathematical statement is true or false.

CLO4- In fact the course inculcates the way thoughts because constructing a legitimate proof involves different skills and expertise

than the discovery part of the process.

CLO5- In this course both angles of problem-solving will be stressed.

 

 

Approach in teaching:

Interactive Lectures, Discussion, Tutorials, Reading assignments, Demonstration, Team teaching

Learning activities for the students:

Self learning assignments, Effective questions, Simulation, Seminar presentation, Giving tasks, Field practical

 

 

 

 

 

 

 

Presentations by Individual Students.

Class Tests at Periodic Intervals.

Written assignment(s)

Semester End Examination

 

 

 

 

 

 

 

 

 

 

Unit I: 
I
15.00

Direct product of groups (external and internal), Isomorphism theorems, Diamond isomorphism theorem, Butterfly lemma, Conjugate classes.

Unit II: 
II
15.00

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

Unit III: 
III
15.00

Modules, Submodules, Quotient modules, Direct sums and module homomorphisms, Generation of modules, Cyclic modules.

Unit IV: 
IV
15.00

Field theory: Extension fields, Algebraic and transcendental extensions, Separable and inseparable extensions, Normal extensions,Splitting fields.

Unit V: 
V
15.00

Galois theory: Elements of Galois theory, Fundamental theorem of Galois theory, Solvability by radicals.   

Essential Readings: 
  • Dileep S.Chauhan and K.N.Singh,Studies in Algebra, JPH, Jaipur, 2011.
  • P.B.Bhattacharya, S.K.Jain, S.R.Nagpaul,Basic Abstract Algebra, Cambridge       University Press,1995.
  • I.N.Herstein, Topics in Algebra, Wiley Eastern Ltd., New Delhi,1975.
  • Knapp, W.Anthony,Advanced Algebra,Springer,2008.
References: 
  • Deepak Chatterjee,Abstract Algebra, PHI. Ltd. New Delhi, 2015.

  • John B.Fraleigh,A First Course in Abstract Algebra,  Narosa Publishing House, New Delhi,2002.

  • S.David, Richard M.Foote Dummit,Abstract Algebra, John Wiely & Sons Inc. USA,2003.
  • S.Hang, Algebra, Addison Wesley, 1993.
  • N.Jacobson, Basic Algebra, Hindustan Publishing Co, 1988.
  • M.Artin, Algebra,Prentice Hall India, 1991.
  • C.Musili, Introduction to Rings and Modules, Narosa Publishing House, NewDelhi, 1994.
Academic Year: 
2019-2020
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