Algebraic Number Theory Course | NPTEL | Prof. Mahesh Kakde IISc
Course Details
| Exam Registration | 18 |
|---|---|
| Course Status | Ongoing |
| Course Type | Elective |
| Language | English |
| Duration | 12 weeks |
| Categories | Mathematics, Algebra |
| Credit Points | 3 |
| Level | Postgraduate |
| 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 |
Unlocking the Mysteries of Numbers: A Deep Dive into Algebraic Number Theory
Algebraic Number Theory stands as one of the most profound and beautiful branches of mathematics, bridging classical number theory with abstract algebra. It provides the essential language and tools for solving ancient Diophantine problems—polynomial equations with integer or rational solutions—by extending our view beyond the familiar integers. This 12-week postgraduate course, offered by the distinguished Prof. Mahesh Kakde of the Indian Institute of Science (IISc) Bangalore, is designed to guide you through this fascinating landscape.
About the Instructor: Prof. Mahesh Kakde
Prof. Mahesh Kakde is a leading expert in the field, having joined IISc Bangalore as a professor in August 2019. With a rich academic background that includes a PhD from Cambridge University under the guidance of John Coates, postdoctoral positions at Princeton University and University College London, and eight years as a faculty member at King’s College London, Prof. Kakde brings a world-class perspective to the subject. His research specializes in Iwasawa theory, focusing on deep conjectures that connect special values of L-functions to fundamental arithmetic objects.
Course Overview and Objectives
The central theme of the course is the study of number fields—finite extensions of the rational numbers obtained by adjoining roots of polynomial equations. Why is this powerful? Consider the simple equation x² + y² = z². By introducing the imaginary unit i (where i² = -1), we can factor it as (x+iy)(x-iy), revealing structure hidden in the integers. Algebraic Number Theory systematizes this approach.
Over 12 weeks, you will master the core objects and invariants that form the backbone of the theory:
- Ring of integers and its units
- Discriminant and Different
- Ideal class group (measuring the failure of unique factorization)
- L-functions and zeta functions
The course blends theoretical foundations with computational techniques, enabling you to not only prove key theorems but also compute these invariants for concrete examples, with direct applications to solving Diophantine equations.
Who Should Enroll?
This course is intended for Master's and PhD students in Mathematics. A solid foundation in Algebra and Galois Theory is a prerequisite. NPTEL offers excellent preparatory courses which are recommended:
- Introduction to Group Theory
- Linear Algebra
- Galois Theory
Industry Support: The techniques and concepts taught have significant applications in modern cryptography, making the knowledge highly relevant beyond pure academia.
Detailed 12-Week Course Layout
| Week | Topics Covered |
|---|---|
| 1 | Number fields, ring of integers, norm and trace. |
| 2 | Absolute & relative discriminant, computing the ring of integers. |
| 3 | Dedekind domains, factorization of prime ideals in extensions. |
| 4 | Embeddings, geometry of numbers, finiteness of class groups. |
| 5 | Computation of class groups and applications to Diophantine equations. |
| 6 | Dirichlet’s Unit Theorem. |
| 7 | Extension of ideals, maps between class groups, decomposition & inertia groups, Frobenius. |
| 8 | Valuations, local fields, Hensel’s Lemma. |
| 9 | Extensions of local fields, ramification, different. |
| 10 | Study of special fields: quadratic, cubic, and cyclotomic fields. |
| 11 | Ray class fields, introduction to class field theory. |
| 12 | Zeta functions, L-functions, analytic properties, Dirichlet’s Class Number Formula. |
Recommended Textbooks
To complement the lectures, Prof. Kakde recommends several authoritative texts:
- Number Fields by Daniel Marcus
- A Brief Guide to Algebraic Number Theory by Peter Swinnerton-Dyer
- Algebraic Number Theory by Jürgen Neukirch
Conclusion
This course offers a rigorous and comprehensive journey into the heart of modern number theory. Guided by an expert in the field, you will gain the tools to explore the intricate relationship between numbers, algebra, and analysis. Whether your goal is to pursue advanced research in areas like Iwasawa theory or to apply these concepts in fields like cryptography, this course provides an indispensable foundation. Enroll to begin your exploration of the elegant and powerful world of Algebraic Number Theory.
Enroll Now →