Advanced Algebra

Paper Code: 
24DMAT711
Credits: 
6
Contact Hours: 
90.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to –

  1. Demonstrate knowledge of conjugacy relations and class equations.
  2. Compute the irreducibility of polynomials.
  3. Develop the concepts of extension fields.
  4. Explain the splitting field for a given polynomial.

 

Course Outcomes: 

 Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

 

 

 

24DMAT

711

 

 

 

 

 

 

Advanced Algebra (Theory)

 

 

 

 

 

 

CO157: Create group, subgroup and direct product of groups, related properties and theorems.

CO158: Differentiate between derived subgroup, solvable group, quotient group and normal subgroup.

CO159: Explain modules, submodules, related properties and theorems and their uses in security systems.

CO160: Analyze extensions of fields and their applications in real life problems.

CO161: Apply the knowledge of Galois field, sub-field, related properties and theorems in encryption and description.

CO162: Contribute effectively in course-specific interaction.

Approach in teaching:

Interactive Lectures, Discussion, Power Point Presentations, Informative videos

 

Learning activities for the students:

Self learning assignments, Effective questions, presentations.

 

 

Quiz, Individual and group projects,

Open Book Test, Semester End Examination

 

 

Unit I: 
Groups
18.00

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

Unit II: 
Series in Groups
18.00

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

Unit III: 
Modules
18.00

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

 

Unit IV: 
Field theory
18.00

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

 

Unit V: 
Galois Theory
18.00

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.
  • John B. Fraleigh, A First Course in Abstract Algebra,  Narosa Publishing House, New Delhi, 2013.
  • 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, 2006.
  • Thomas W. Hungerford, Algebra (Graduate Texts in Mathematics), Springer, 1975.

 

References: 
  • Deepak Chatterjee, Abstract Algebra, PHI. Ltd. New Delhi, 2015.
  • 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, New Delhi, 1997.
  • Knapp, W. Anthony, Advanced Algebra, Springer, 2008.

e- RESOURCES

JOURNALS

●     https://www.journals.elsevier.com/journal-of-algebra

●     https://www.worldscientific.com/worldscinet/jaa

 

Academic Year: 
2024-2025
  • Printer-friendly version
  • PDF version