Introduction to Number Theory Course | IIT Kanpur | Prof. Somnath Jha
Course Details
| Exam Registration | 61 |
|---|---|
| Course Status | Ongoing |
| Course Type | Elective |
| Language | English |
| Duration | 12 weeks |
| Categories | Mathematics |
| Credit Points | 3 |
| Level | Undergraduate |
| Start Date | 19 Jan 2026 |
| End Date | 10 Apr 2026 |
| Enrollment Ends | 02 Feb 2026 |
| Exam Registration Ends | 20 Feb 2026 |
| Exam Date | 26 Apr 2026 IST |
| NCrF Level | 4.5 — 8.0 |
An Introduction to Number Theory: A 12-Week Journey into the Heart of Mathematics
Welcome to a fascinating exploration of one of the oldest and most profound branches of mathematics: Number Theory. This 12-week undergraduate course, designed and taught by Prof. Somnath Jha of IIT Kanpur, offers a structured pathway into the elegant world of integers, prime ideals, and Diophantine equations. Whether you're a mathematics student seeking to deepen your understanding or a professional looking to grasp the foundations of modern cryptography, this course provides the essential toolkit.
Meet Your Instructor: Prof. Somnath Jha
Leading this intellectual journey is Prof. Somnath Jha, an accomplished mathematician with a rich academic background. He earned his PhD from the prestigious Tata Institute of Fundamental Research (TIFR), Mumbai, in 2012. His research career includes postdoctoral positions as a MATCH researcher at the University of Heidelberg, Germany, and as a JSPS fellow at Osaka University, Japan. Currently, as an Associate Professor in the Department of Mathematics and Statistics at IIT Kanpur, his research focuses on Number Theory and Arithmetic Geometry, with specific interests in Iwasawa theory, elliptic curves, modular forms, and classical Diophantine problems like congruent numbers and rational cube sums.
Course Overview and Objectives
This course is meticulously crafted to introduce fundamental concepts in number theory, bridging classical ideas with modern applications. The primary objectives are:
- To understand the structure of quadratic number fields, including their rings of integers, prime ideals, and units.
- To explore the failure of unique factorization using the classical theory of binary quadratic forms.
- To investigate famous Diophantine problems, such as those related to congruent numbers and rational cube sums, and discover their deep connections with elliptic curves.
- To review necessary concepts from abstract algebra, ensuring a solid foundation for all participants.
Who Should Enroll?
This course is ideally suited for:
- Undergraduate students pursuing BS, BSc, or BMATH degrees.
- Postgraduate students in MSc or MMATH programs.
- Anyone with a keen interest in pure mathematics and its applications.
Prerequisite: A basic familiarity with linear algebra will be beneficial for following the lectures smoothly.
Industry Relevance: Cryptography and Cybersecurity
The concepts taught in this course are not just theoretical curiosities; they form the bedrock of modern Cryptography and Cybersecurity. The hardness of problems like integer factorization (central to RSA encryption) and the discrete logarithm problem (used in elliptic curve cryptography) are deeply rooted in the number theory principles covered in this curriculum. Understanding these foundations is crucial for professionals and researchers in the field of digital security.
Detailed 12-Week Course Layout
| Week | Topics Covered |
|---|---|
| Week 1 | Abelian groups, subgroups, quotient groups, finite and finitely generated abelian groups. |
| Week 2 | Commutative Rings, ideals, fields, polynomial rings, zero sets of ideals. |
| Week 3 | The groups Z/nZ and (Z/nZ)*, Euler’s theorem, Wilson's Theorem, Chinese Remainder Theorem. |
| Week 4 | Classification of finite fields, structure of their multiplicative subgroups. |
| Week 5 | Law of quadratic reciprocity, Quadratic fields, their containment inside cyclotomic fields. |
| Week 6 | Unique Factorization Domains (UFD) and Principal Ideal Domains (PID), Ring of integers of quadratic fields. |
| Week 7 | Binary quadratic forms. |
| Week 8 | Ideal class groups of imaginary quadratic fields. |
| Week 9 | Units in rings of integers, Diophantine problem: Bramhagupta-Pell’s equation. |
| Week 10 | Infinite descent, n=4 case of Fermat’s Last Theorem, Congruent number problem and its relation to elliptic curves. |
| Week 11 | Diophantine problem on rational cube sums and its relation to elliptic curves. |
| Week 12 | Group law on elliptic curves, statements of the Nagell-Lutz Theorem and the Mordell-Weil Theorem. |
Essential Reading and Reference Materials
To complement the lectures, Prof. Jha recommends a selection of classic and modern texts:
- A Friendly Introduction to Number Theory by J. H. Silverman (Pearson Prentice Hall, 2006)
- A Classical Introduction to Modern Number Theory by K. Ireland & M. Rosen (Springer GTM)
- An Introduction to the Theory of Numbers by I. Niven, H. S. Zuckerman, & H. L. Montgomery (Wiley)
- An Introduction to the Theory of Numbers by G. H. Hardy & E. M. Wright (Oxford)
- Abstract Algebra by D. S. Dummit & R. M. Foote (Wiley)
- Rational Points on Elliptic Curves by J. H. Silverman & J. Tate (Springer UTM)
- Introduction to Elliptic Curves and Modular Forms by N. Koblitz (Springer-Verlag GTM)
- A Course in Arithmetic by J. P. Serre (Springer-Verlag GTM)
Embark on this 12-week journey to unravel the secrets of numbers, from ancient problems to the mathematics securing your online transactions today. Under the expert guidance of Prof. Somnath Jha, you will gain not just knowledge, but a profound appreciation for the beauty and utility of number theory.
Enroll Now →