Advanced Algebra

Paper Code: 
DMAT 701
Credits: 
6
Contact Hours: 
90.00
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 outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

DMAT 701

 

 

 

 

 

Advanced Algebra (Theory)

 

 

 

 

 

 

 

 

The students will be able to –

 

CO75: Describe group, subgroup, direct product of groups, related properties and theorems.

CO76: Differentiate between derived subgroup, solvable group, quotient group and normal subgroup.

CO77: Identify homomorphism and isomorphism theorems of groups.

CO78: Describe modules, submodules, related properties and theorems and their uses in security systems.

CO79: Use the knowledge of Galois field, sub-field, related properties and theorems in encryption and description.

CO80: Analyse extensions of fields and their applications in real life problems.

Approach in teaching:

 

Interactive Lectures, Discussion, Power Point Presentations, Informative videos

 

Learning activities for the students:

Self learning assignments, Effective questions, presentations, Giving tasks

 

Quiz, Poster Presentations,

Power Point Presentations, Individual and group projects,

Open Book Test, Semester End Examination

 

 

 

 

Unit I: 
I
18.00

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

Unit II: 
II
18.00

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

Unit III: 
III
18.00

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

Unit IV: 
IV
18.00

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

Unit V: 
V
18.00

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 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.
Academic Year: 
2023-2024
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