Algebra

By Dr.T.Asir | Pondicherry University

Learners enrolled: 808

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.

Summary

Course Status :	Upcoming
Course Type :	Core
Language for course content :	English
Duration :	15 weeks
Category :	Mathematics
Credit Points :	5
Level :	Undergraduate
Start Date :	09 Jan 2025
End Date :	30 Apr 2025
Enrollment Ends :	28 Feb 2025
Exam Date :	17 May 2025 IST
NCrF Level	4.5
Exam Shift :	Shift - 1

Note: This exam date is subject to change based on seat availability. You can check final exam date on your hall ticket.

Page Visits

Course layout

Week – 1

1. Polar Representation of Complex Numbers

2. nth Root of Unity

3. De Moivre’s Theorem for Rational Indices

4. Equivalence Relation

Week – 2

5. Composition of Functions and Invertible Functions

6. One-to-one Correspondence and Cardinality of a set

7. Division Algorithm

8. Divisors and Euclidean Algorithm

Week – 3

9. Congruence Relation Between Integers

10. Statement of Fundamental Theorem of Arithmetic and Principle of Mathematical Induction

11. Systems of Linear Equations, Row Reduction and Echelon Forms

12. The Matrix Equation Ax=b and the Solution Set of Linear System

Week – 4

13. Application Of Linear System and Linear Independence

14. Introduction to linear transformation and matrix of a linear transformation

15. The Inverse of a Matrix

16. Subspace of ln and Rank of a matrix

Week – 5

17. Eigen value and Eigen vector - I

18. Eigen value and Eigen vector - II

19. Diagonalization of Matrix

20. Problems

Week – 6

21. Groups Introduction - I

22. Groups Introduction - II

23. Subgroups

Week – 7

24. Elementary Properties of Abelian Groups

25. Elementary Properties of Non-Abelian Groups

26. The Group of Integer Modulo n

Week – 8

27. Complex Roots of Unity

28. Cyclic Group

29. General Linear Group

Week – 9

30. The Group of Symmetries

31. Subgroup Generated by a Subset

32. Cosets and Index of Subgroups

Week – 10

33. Properties of Group Homomorphisms

34. Normal Subgroups

35. Quotient Groups

Week – 11

36. Class Equation

37. Direct Product of a Finite Number of Groups

38. Fundamental Theorem of Finite Abelian Groups

39. Cayley Theorem and Problems in Group Theory

Week – 12

40. Ring-Introduction

41. Class of Ring

42. Rings from Number Systems

Week – 13

43. Ring of Real Quaternions and Rings of Continuous Functions

44. Rings of Matrices

45. Polynomial Rings

46. Subrings and Ideals

Week – 14

47. Set of Zero Divisors and Group of Units

48. Properties of Ring Homomorphisms

49. Operations on Ideals

Week – 15

50. Integral Domains and Fields

51. Maximal Ideals and Prime Ideals

52. Euclidian Domains and Principal Ideal Domains

53. Unique Factorization Domains

Books and references

1. Titu Andreescu and Dorin Andrica, Complex Numbers from A to Z, Birkhauser, 2006.

2. Edgar G. Goodaire and Michael M. Parmenter, Discrete Mathematics with Graph Theory, 3rd Ed., Pearson Education (Singapore) P. Ltd., Indian Reprint, 2005.

3. David C. Lay, Linear Algebra and its Applications, 3rd Ed., Pearson Education Asia, Indian Reprint, 2007.

4. Joseph A. Gallian, Contemporary Abstract Algebra, 4th Ed., Narosa Publishing House, New Delhi, 1999.

5. I.N. Herstein, Topics in Algebra, Wiley Eastern Limited, India, 1975.

6. David S. Dummit and Richard M. Foote, Abstract Algebra, 3rd Ed., John Wiley and Sons(Asia) Pvt. Ltd, Singapore, 2004.

Instructor bio

Dr.T.Asir

Pondicherry University

Dr. T. Asir has received a gold medal in M.Sc. and secured University second rank in M.Phil. He received the Ph.D. degree in Mathematics at Manonmaniam Sundaranar University and his Ph.D. work was supported by the grants “Major Project Fellowship” by University Grants Commission(UGC), India and the “INSPIRE Fellowship” by the Department of Science & Technology(DST), India. He did his Post Doctorate under the UGC-Kothari Postdoctoral Fellowship. He worked as an Assistant Professor in Department of Mathematics-DDE, Madurai Kamaraj University, Tamil Nadu from Dec 2013 to July 2022.

His research interest is Algebra and Graph Theory. More specifically, graphs arising from rings and vice versa. He has published 36 research articles in International Journals (Journals like Discrete Mathematics, Discrete Applied Mathematics, Expositiones Mathematicae, Communications in Algebra, Soft Computing) and two books (including a book in Springer) with a citation count of 256 on Web of Science, 350 on Scopus and 545 on Google Scholar. He has completed three projects UGC-Startup grant, SERB-MATRICS project and Deanship of Scientific Research at King Abdulaziz University, Saudi Arabia. Currently, his research is supported by CSIR project.

Further 5 Ph.D.’s and 25 M.Phil.’s have been awarded under his guidance. Moreover, he has delivered more than 75 invited talks at various National/International conferences and Orientation/Refreshers programs. He has served as a referee of several international journals in the frame of pure and applied mathematics. Furthermore, he has developed online mathematical content through MOOC Courses “Core and Pedagogy of Mathematics”, “Modern Algebra” and “Graph Theory” in SWAYAM Platform (MHRD).