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. Identify the irreducibility of polynomials.
  3. Develop the concepts of extension fields. 
  4. Find the splitting field for a given polynomial.

 

Course Outcomes (COs):

course learning

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

 

 

 

MAT 121

 

 

Advanced Algebra (Theory)

 

 

 

 

 

 

The students will be able to –

 

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

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

CO3: Identify homomorphism and isomorphism theorems of groups.

CO4: Describe modules, submodules, related properties, theorems, and their uses in security systems.

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

CO6: Analyse extensions of fields and their applications in real-life problems.

 

The approach to teaching:

 

Interactive Lectures, discussions, PowerPoint Presentations, Informative videos

 

Learning activities for the students:

Self-learning assignments, Effective questions, presentations, Field trips

 

 

Quiz, Poster Presentations,

PowerPoint Presentations, Individual and group projects,

Open Book Test, 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.
  • 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 Wiley & 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.
Academic Year: 
2022-2023
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