Advanced Algebra

Paper Code: 
25MAT121
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. Identify the irreducibility of polynomials.
  3. Develop the concepts of extension fields. 
  4. Find 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

25MAT 121

 

 

 

Advanced Algebra (Theory)

 

 

 

 

 

 

CO1: Create group, subgroup and direct product of groups, related properties and theorems.

CO2: Differentiate between derived subgroup, solvable group, quotient group and normal subgroup.

CO3: Explain modules, sub-modules, related properties and theorems and their uses in security systems.

CO4: Analyze extensions of fields and their applications in real life problems.

CO5: Apply the knowledge of Galois field, sub-field, related properties and theorems in encryption and description.

CO6: Contribute effectively in course-specific interaction.

Approach in teaching:

Interactive Lectures, Discussion, Informative videos

 

Learning activities for the students:

Self learning assignments, Effective questions,  Topic  presentation, Assigned tasks

 

 

 

 

Quiz, Class Test, Individual projects,

Open Book Test, Continuous Assessment, Semester End Examination

 

 

 

 

 

 

Unit I: 
Groups:
15.00
Direct product of groups (external and internal), Isomorphism theorems, Diamond isomorphism theorem, Butterfly lemma, Conjugate classes.
 
Unit II: 
Series in Groups:
15.00
Commutators, Derived subgroups, Normal series and solvable groups, Composition series, Refinement theorem and Jordan-Holder theorem for infinite groups.
 
Unit III: 
Modules:
15.00
Modules: Modules, Submodules, Quotient modules, Direct sums and module homomorphisms, Generation of modules, Cyclic modules.
 
Unit IV: 
Field theory:
15.00
Extension fields, Algebraic and transcendental extensions, Separable and inseparable extensions, Normal extensions, Splitting fields.
 
Unit V: 
Galois Theory:
15.00

 Elements of Galois Theory, Fundamental theorem of Galois Theory, Solvability by radicals.

Essential Readings: 
ESSENTIAL READING
  • 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: 
SUGGESTED READING
  • Deepak Chatterjee, Abstract Algebra, PHI. Ltd. New Delhi, 2015.
  • David S. Dummit, Richard M. Foote, 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 
 
Academic Year: 
2025-2026
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