Course Content: 

 

  1. Basic concepts from set theory, definition of groups, subgroups,
  2. Lagrange’s theorem, normal subgroups, homomorphisms, factor groups, theorems concerning homomorphisms. (4 Lectures)
  3. Group actions, Cayley’s theorem, Orbit-stabilizer theorem, conjugacy classes and class equation, Sylow’s theorems. (10 Lectures)
  4. Free groups, generators and presentation of groups, Direct and semi-direct product of groups. (3 Lectures)
  5. Definition of Rings, commutative rings, ideals, prime and maximal ideals, existence of maximal ideals, quotient construction, isomorphism theorems, Chinese remainder theorem, fraction fields of domains. (17 Lectures)
  6. Principal ideal domains, Euclidean domains, unique factorisation domains, Gauss lemma, polynomial rings and irreducibility criteria. (8 Lectures)

 

Learning Outcomes: upon successful completion of the course,

  1. the students will have a good understanding of the theory of groups and rings. 
  2. they will be able to appreciate the power of abstraction and gain mathematical maturity.

 

Text Books: 

  1.  Michael Artin, Algebra, Pearson India Education Services Pvt.Ltd, ISBN: 978-93-325-4983-8       
  2.  IN Herstein, Topics in algebra, Wiley india Pvt.Ltd, ISBN 978-81-265-1018-4
  3. John B Fraleigh, A first course in abstract algebra, Addison-wesley, ISBN : 978-02-015-3467-2
  4.  N. Jacobson, Basic Algebra I, Dover publications, ISBN-10 : 9780486471891    
  5. David S. Dummit, Richard M. Foote, Abstract Algebra,  Wiley india Pvt.Ltd, ISBN-10 8126532289

 

References

  1. N. Bourbaki, Algebra I, Springer-Verlag Berlin Heidelberg, ISBN 978-3-540-64243-5
  2. Serge Lang, Algebra, volume 211 of Graduate Texts in Mathematics, Springer-Verlag New York, ISBN : 978-0-387-95385-4, 978-1-4612-6551-1