NPTEL course on Algebraic Number Theory
I am teaching an eight week NPTEL course on Algebraic Number Theory, with a focus on the commutative algebraic aspects of it during the July-December 2026 semester. It is targeted at Master's students in India, interested in pursuing research in Number theory. It is titled "Commutative Algebra with a viewpoint towards Algebraic Number Theory" and is a deliberate nod to the title of David Eisenbud's book.
The lecture videos are uploaded on Youtube and can be found as a YouTube playlist. Details about the course (including course notes, homework assignments, exam registration etc) can be found on NPTEL and Swayam link.
The course TAs are Ravitheja Vangala and Sohan Ghosh.
Some important dates:
- Course dates: 17 August - 09 October 2026
- Exam date: 24 October 2026
Here's the course outline:
- Week 1: Integral extensions and going up theorems
(References: Lang, Chapter I and Atiyah--Macdonald, Chapter 5) - Lecture 1: Integral elements
- Lecture 2: Integral closures
- Lecture 3: Integral closures (contd), Trace and Norm
- Lecture 4: Integral closure of a Noetherian integrally closed domain
- Lecture 5: Rings of algebraic integers of global fields
- Lecture 6: Going up (lying above theorem)
- Week 2: Valuation Rings (References: Atiyah--Macdonald, Chapter 5 and Stacks Project)
- Lecture 7: Local rings and localizations
- Lecture 8: V-rings and V*-rings
- Lecture 9: Valuations and Valuation ring
- Lecture 10: V-rings are valuation rings
- Lecture 11: Maximal local rings inside fields
- Lecture 12: Maximal local rings and valuation rings
- Lecture 13: Characterization of interally closed domains involving valuation rings
- Week 3: Discrete Valuation Rings (References: Atiyah--Macdonald, Chapter 9 and Cassels--Fröhlich, Chapter 1)
- Lecture 14: Discrete Valuation Rings - Part 1
- Lecture 15: Discrete Valuation Rings - Part 2
- Lecture 16: Classification of DVRs inside Z and Fp[t]
- Week 4: $p$-adic integers and Hensel's lemma (Reference: Lang, Chapter II)
- Lecture 17: p-adic absolute values
- Lecture 18: Q as a topological field
- Lecture 19: p-adic numbers
- Lecture 20: p-adic integers
- Lecture 21: p-adic integers as a completion
- Lecture 22: p-adic expansions of p-adic integers
- Lecture 23: Hensel's lemma
- Week 5: Dedekind domains (References: Lang Chapter I, Cassels--Fröhlich Chapter 1, Atiyah--Macdonald Chapter 9)
- Week 6: Galois extensions (Reference: Lang Chapter I)
- Week 7: Class groups
- Week 8: Finiteness of Class groups (Reference: Alexander Stasinski's article in the American Mathemtical Monthly:
A Uniform Proof of the Finiteness of the Class Group of a Global Field).
Bharathwaj Palvannan
Assistant Professor, 2021-
X04, Department of Mathematics,
Indian Institute of Science, Bangalore
bharathwaj [at] iisc [dot] ac [in]
Research interests:
Number theory
CV