Algebra

The course on Algebra is a 5 credit course for the Undergraduate programme. In this course, we have given basic algebraic structure required to understand Modern Algebra. Also we have covered the basic concepts of group theory and ring theory as extensively as possible. This course will serve as a useful tool to any learner who wishes to learn Algebra, Linear Algebra, Topology and Algebraic Number Theory. Also, this course is essential for all learners planning for advance degree in mathematics or planning to enter the teaching profession.

Week – 1

1. Polar Representation of Complex Numbers2. nth Root of Unity3. De Moivre’s Theorem for Rational Indices4. Equivalence Relation
Week – 2
5. Composition of Functions and Invertible Functions6. One-to-one Correspondence and Cardinality of a set7. The Division Algorithm8. Divisors and Euclidean Algorithm
Week – 3
9. Congruence Relation Between Integers10. Statement of Fundamental Theorem of Arithmetic and Principle of Mathematical Induction11. Systems of Linear Equations, Row Reduction and Echelon Form12. The Matrix Equation Ax and the Solution Set
Week – 4
13. Application Of Linear System and Linear Independence14. Introduction to linear transformation and matrix of a linear transformation15. Invertible Matrices16. Subspace of Rn and Rank
Week – 5
17. Eigen value and Eigen vector - I18. Eigen value and Eigen vector - II19. Diagonalization of matrix20. Problems
Week – 6
21. Groups Introduction - I22. Groups Introduction - II23. Subgroups
Week – 7
24. Elementary properties of abelian groups25. Elementary properties of non-abelian groups26. The group of integer modulo n
Week – 8
27. Complex roots of unity28. Cyclic group29. The general linear group
Week – 9
30. The Group of symmetries31. The subgroup generated by a subset32. Cosets and Index of subgroup
Week – 10
33. Properties of group homomorphisms 34. Normal subgroups35. Quotient groups
Week – 11
36. Class Equations37. Direct product of a finite number of groups38. Fundamental Theorem of Finite Abelian Groups39. Cayley Theorem and Problems in Group Theory
Week – 12
40. Ring-Introduction41. Class of Ring42. Rings from number systems
Week – 13
43. Ring of real quaternions & Rings of continuous functions44. Rings of matrices45. Polynomial rings46. Subrings and ideals
Week – 14
47. Set of zero Divisors and group of units48. Properties of Ring homomorphisms49. Operations on ideals
Week – 15

50. Integral domains and fields51. Maximal ideals and Prime ideals52. Euclidian Domains and Principal Ideal Domains53. Unique factorization Domains