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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

Topics covered: Definition of an integral element, Equivalent criteria for an element to be integral, Definition of an integral extension.
Lecture 2: Integral closures

Topics covered: Review of basic facts about finitely generated algebras and modules over a ring, Definition of ring of integers, Definition of integral closures, Integral extensions is a transitive property, Integral closure is integrally closed.
Lecture 3: Integral closures (contd), Trace and Norm

Topics covered: "Clearing denominator trick" to get integral element, Definition and properties of Noetherian rings and modules, Basic facts from field and Galois theory, Definition and properties of trace and norm, Properties of trace and norm of an element in a finite separable extension.
Lecture 4: Integral closure of a Noetherian integrally closed domain

Topics covered: Characterisation of integral closures of Noetherian integrally closed domains over finite separable extensions, Proof of Proposition 8 of Lecture-3: integral closure of a Noetherian integrally closed domain in a finite separable field extension is a finitely generated module.
Lecture 5: Rings of algebraic integers of global fields

Topics covered: Definition of UFD and PID, Review of basic facts about UFD and PID, UFD is integrally closed, Integral closure of PID in a finite seperable extension
Lecture 6: Going up (lying above theorem)

Topics covered: Lying above property holds for an integral extension

Week 2: Valuation Rings

Stacks Project

Lecture 7: Local rings and localizations

Topics covered: Definition of local ring, Definition of multiplicatively closed set, Definition of localization, Ideals in localization, Localization at a prime is local, Localization of a module, Localization of integral extension is integral, Localization of integral closure is again a integral closure
Lecture 8: V-rings and V*-rings

Topics covered: Definition of local ring, Definition of multiplicatively closed set, Definition of localization, Ideals in localization, Localization at a prime is local, Localization of a module, Localization of integral extension is integral, Localization of integral closure is again a integral closure
Lecture 9: Valuations and Valuation ring

Topics covered: Definition of a totally ordered abelian group, a totally ordered abelian group is torsion-free. Definition of valuation, ultramteric property, properties of valuation. Valuation ring is a subring of K, valuation ring is a V-ring
Lecture 10: V-rings are valuation rings

Topics covered: Construction of a totally ordered abelian group associated to a V-ring, Construction of a valaution asscotiated to a V-ring, Proof that Valuation rings are V-rings
Lecture 11: Maximal local rings inside fields
Topics covered: Definition of map of local rings, Definition of domination of local rings and maximal under domination inside a field, Valuation ring is maximal under domination, Ring maximal under domination inside a field is integrally closed.
Lecture 12: Maximal local rings and valuation rings
Topics covered: A local ring maximal under domination is a valuation ring.
Lecture 13: Characterization of interally closed domains involving valuation rings

Topics covered: Equivalence of V-ring, V*-ring, Valuation ring and maximal local rings inside field of fractions, Existence of valuation ring dominating a local ring contained a field, An integrally closed domain is the intersection of valuation rings

Week 3: Discrete Valuation Rings

Lecture 14: Discrete Valuation Rings - Part 1

Topics covered: Recall of totally ordered abelian group, Definition of isomorphism of totally ordered abelian group, Recall of definition of valuation, Image of valution is a totally ordered abelian group, Definition value group, Definition of Discrete valuation ring, Equivalent conditions of a valuation ring with value group ℤ
Lecture 15: Discrete Valuation Rings - Part 2

Topics covered: Proof of Equivalent conditions of DVR continued from Lecture-14
Lecture 16: Classification of DVRs inside Z and Fp[t]

Topics covered: Localization of PID at a prime is a discrete valuation ring, Valuation ring containing PID inside the field of fractions is given by localization at a prime, Classification of valuation subrings of ℚ containing ℤ, Classification of valuation rings containing polynomial ring with coefficients in finite field (𝔽p[t])

Week 4: $p$-adic integers and Hensel's lemma

Lecture 17: p-adic absolute values

Topics covered: Definition p-adic valuation and p-adic absolute value on localization of integers at p, Proving p-adic absolute value satisfies ultrametric inequality and the triangle inequality, Definition of p-adic metric, ℚ is metric space with respect to p-adic metric, p-adic metric on ℚ×ℚ
Lecture 18: Q as a topological field

Topics covered: Recall of facts from topology regarding Cauchy sequences and Convergent sequences, Addition is a contiuous map on ℚ×ℚ, Definition of null sequence, p-adic absolute value stabilizes to a non-zero number in a non-null Cauchy/Convergent sequence, Inversion maps is conitinuos on non-zero rational, Multiplication map is continuous on units in ℚ×ℚ
Lecture 19: p-adic numbers

Topics covered: Construction of p-adic numbers ℚp, Set of Cauchy sequences (C) in ℚ w.r.t.p-adic norm is a commutative ring, Extending p-adic absolute value to C, Set of null sequences (N) in ℚ w.r.t. p-adic absolute value is a maximal ideal of C
Lecture 20: p-adic integers

Topics covered: ℚ is a subring of ℚp, Extending p-adic valuation to ℚp, Definition of p-adic integers ℤp, Maximal of p-adic integers is generated by p, p-adic numbers is complete w.r.t. p-adic absolute value.
Lecture 21: p-adic integers as a completion

Topics covered: Showing that ℤp is completion of ℤ and ℤ(p) w.r.t. p-adic norm, that is, ℤp is the quotient of Cauchy Sequences in ℤ and ℤ(p) modulo the ideal of null sequences.
Lecture 22: p-adic expansions of p-adic integers

Topics covered: Recall of the construction of ℝ from ℚ, Explicit p-adic expansion of an element of ℤp by finding an optimal Cauchy Sequence in ℤ, Definition of p-adic expansion and p-adic digits, Residue field of p-adic integers is 𝔽p
Lecture 23: Hensel's lemma
Topics covered: Recall of the construction of ℝ from ℚ, Explicit p-adic expansion of an element of ℤp by finding an optimal Cauchy Sequence in ℤ, Definition of p-adic expansion and p-adic digits, Residue field of p-adic integers is 𝔽p

Week 5: Dedekind domains

Lecture 24: Dedekind Domains

Topics covered: Definition of a fractional ideal, Product of two fractional ideals is fractional, Definition of a invertible fractional ideal, Definition of inverse of a fractional ideal, Inverse of a fractional ideal is an A-module, Definition of a Dedekind Domain, Every ideal of a Dedekind domain contains a product of non-zero prime ideals, Every non-zero maximal ideal of a Dedekind domain is invertible.
Lecture 25: Unique Factorisation of Ideals in Dedekind Domains

Topics covered: Every non-zero ideal of a Dedekind Domain is invertible, Every non-zero fractional ideal of a Dedekind Domain is a product of prime ideals, Definition of Class group
Lecture 26: Ring of Integers are Dedekind Domains

Topics covered: Strengthening of lying above theorem, Ring of integers in a finite extension of ℚ is a Dedekind Domain, Ring of integers of finite extension in a Function field is a Dedekind Domain, Structure theorem for f.g. modules over Dedekind Domains, Invariant factor form and elementary fator form for f.g. torsion modules over Dedekind Domains

Week 6: Galois extensions

Lecture 27: Galois extensions

Topics covered: Definition of a normal extension, Definition and examples of Galois extensions, Galois group, Prime ideals lying above a given prime and their characterization via ideal factorization, Action of the Galois group on the integral closure and prime ideals, Finiteness of prime ideals lying above a fixed prime, Chinese Remainder Theorem, Transitive action of the Galois group on the set of prime ideals lying above a fixed prime.
Lecture 28: Decomposition Groups

Topics covered: Decomposition groups of prime ideals in finite Galois extensions, Characterization of decomposition fields via splitting of prime ideals, Minimality property of the decomposition field (fixed field of decomposition group), Proof that the induced map on residue fields is an isomorphism when the intermediate field is the decomposition field of β
Lecture 29: Ramification indices and residue field degrees

Topics coverd: Ramification indices and residue field degrees, Equality of ramification indices and residue field degrees of primes lying above a fixed prime, Normality of residue field extensions, Galois action on residue fields, Natural surjective homomorphism from the decomposition group to the Galois group of the residue field extension, Definition of the inertia group.
Lecture 30: efg theorem

Topics covered: The (efg)-theorem for finite separable extensions of Dedekind domains, Trick of localization, Localization of Dedekind domains and preservation of ramification indices and residue field degrees, Proof of the (efg)-theorem using localization and the Chinese Remainder Theorem.
Lecture 31: Inertia group

Topics covered: Relationship between the decomposition group, inertia group, and ramification index, Computation of the order of the decomposition and inertia groups, Decomposition of the residue field degree into separable and inseparable parts, Characterisation of the inertia group via residue field extensions, Construction of the maximal unramified subextension, Proof that the fixed field of the inertia group is an unramified extension of the decomposition field

Week 7: Class groups
(Reference: Pete Clark's notes)

Lecture 32: Class groups

Topics covered: Picard groups and their relationship with the ideal class group of a Dedekind domain, Invertible modules and invertible fractional ideals, Rank one projective modules, Equivalent characterisation of Projective modules, Equivalence between invertible modules, rank one projective modules, and invertible fractional ideals, Dual basis characterisation of finitely generated projective modules, Equivalence between projective modules and modules admitting a dual basis.
Lecture 33: Equivalent characterisation of invertible fractional ideals

Topics covered: Proof of the equivalence between invertible modules and invertible fractional ideals, Proof that invertible modules are rank-one projective modules, Proof that rank-one projective modules are isomorphic to invertible fractional ideals.

Week 8: Finiteness of Class groups

Alexander Stasinski's

A Uniform Proof of the Finiteness of the Class Group of a Global Field

Lecture 34: Basic PID

Topics covered: Finiteness of a norm of an ideal in Global field setting, Definition of basic PID, ℤ and 𝔽p[t] are basic PID.
Lecture 35: Norm Estimates

Topics covered: Norm of a principal ideal is same as the field norm, Norm of every element is evaluation of a homogeneous fixed polynomial.
Lecture 36: Finiteness of Class group I

Topics covered: Every non-zero ideal I of a Dedekind Domain contains an element with field norm less than C Norm(I) for some fixed constant depending only on the field.
Lecture 37: Finiteness of Class group II

Topics covered: There exists finitely many ideals of bounded norm, Completing the proof of finiteness of class group

NPTEL course on Algebraic Number Theory

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