Course11110800337 lectures
Commutative Algebra with a Viewpoint towards Algebraic Number Theory
Prof. Bharathwaj Palvannan · IISc Bangalore
About
Algebraic number theory involves the study of global fields (number fields and function fields), their associated local fields obtained via completions (such as the p-adic integers), and the arithmetic information encoded in L-functions. Following the approach of classical references such as Cassels– Fröhlich and Lang's Algebraic Number Theory, we develop the necessary commutative algebra — integral extensions, valuation rings, discrete valuation rings, and Dedekind domains — that is an integral part of this subject. The title of the course reflects this perspective and is a deliberate reference to Eisenbud’s Commutative Algebra with a View Toward Algebraic Geometry.
The two central theorems of a standard algebraic number theory course are the finiteness of class groups and Dirichlet's unit theorem, both of which classically require Minkowski's geometry of numbers. In this 8- week course, we omit Dirichlet's unit theorem and instead present the finiteness of class groups via a more algebraic approach.
Prerequisites: Rings and modules, Commutative Algebra (Localization and Noetherian rings), Galois theory
Lectures
Week 1 Integral extensions
- Integral Element
- Integral Closure
- Integral Closure, Trace and Norm
- Integral closure of a Noetherian integrally closed domain
- Ring of Algebraic Integers
- Going up theorem
Week 2 Valuation rings
- Local rings and localizations
- V and V* rings
- Valuation and Valuation ring
- V-rings are valuation rings
- Maximal local rings inside fields
- Maximal local rings and Valuation rings
- A characterization of integrally closed domains involving valuation rings
Week 3 Discrete Valuation Rings
- Discrete Valuation rings-1
- Discrete Valuation rings-2
- Classification of Discrete Valuation rings containing ℤ and Fp[t]
Week 4 Completions and Hensel's lemma
- p-adic absolute value
- ℚ is a topological field
- p-adic numbers
- p-adic integers
- p-adic integers as a completion
- p-adic expansion of p-adic integers
- Hensel's lemma
Week 5 Dedekind Domains
- Dedekind Domains
- Unique Factorisation of Ideals in Dedekind Domains
- Ring of Integers are Dedekind Domains
Week 6 Galois extensions of global fields
- Galois extensions
- Decomposition Groups
- Ramification indices and residue field degrees
- efg theorem
- Inertia group
Week 7 Class groups
Week 8 Finiteness of Class group
Syllabus and materials
At NPTEL, as the course publishes them.
Credits and licence
- Institution
- IISc Bangalore
- Department
- Mathematics
- Teacher
- Prof. Bharathwaj Palvannan
- Level
- Graduate
- Language
- English
- Subjects
- Mathematics
- Source
- NPTEL
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
- Who can watch
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