# Commutative Algebra with a Viewpoint towards Algebraic Number Theory

- Institution: IISc Bangalore
- Department: Mathematics
- Teacher: Prof. Bharathwaj Palvannan
- Level: Graduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111108003)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111108003)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

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

1. [Integral Element](https://invisiblenetwork.ai/learn/nptel/111108003/0fc01e4f4442) (YouTube)
2. [Integral Closure](https://invisiblenetwork.ai/learn/nptel/111108003/251bc564c80f) (YouTube)
3. [Integral Closure, Trace and Norm](https://invisiblenetwork.ai/learn/nptel/111108003/be84f6df21e1) (YouTube)
4. [Integral closure of a Noetherian integrally closed domain](https://invisiblenetwork.ai/learn/nptel/111108003/6894f1f16289) (YouTube)
5. [Ring of Algebraic Integers](https://invisiblenetwork.ai/learn/nptel/111108003/c370bc2fc006) (YouTube)
6. [Going up theorem](https://invisiblenetwork.ai/learn/nptel/111108003/8f6094578546) (YouTube)
7. [Local rings and localizations](https://invisiblenetwork.ai/learn/nptel/111108003/39c1443984e6) (YouTube)
8. [V and V* rings](https://invisiblenetwork.ai/learn/nptel/111108003/bdf6c29f2a80) (YouTube)
9. [Valuation and Valuation ring](https://invisiblenetwork.ai/learn/nptel/111108003/d2c5b839cc7b) (YouTube)
10. [V-rings are valuation rings](https://invisiblenetwork.ai/learn/nptel/111108003/0df2d793ca89) (YouTube)
11. [Maximal local rings inside fields](https://invisiblenetwork.ai/learn/nptel/111108003/32fa0363c449) (YouTube)
12. [Maximal local rings and Valuation rings](https://invisiblenetwork.ai/learn/nptel/111108003/ca3b6c3855c0) (YouTube)
13. [A characterization of integrally closed domains involving valuation rings](https://invisiblenetwork.ai/learn/nptel/111108003/8b5520c8cbf7) (YouTube)
14. [Discrete Valuation rings-1](https://invisiblenetwork.ai/learn/nptel/111108003/c90e87814d6b) (YouTube)
15. [Discrete Valuation rings-2](https://invisiblenetwork.ai/learn/nptel/111108003/901044521d59) (YouTube)
16. [Classification of Discrete Valuation rings containing ℤ and Fp\[t\]](https://invisiblenetwork.ai/learn/nptel/111108003/2127709ba748) (YouTube)
17. [p-adic absolute value](https://invisiblenetwork.ai/learn/nptel/111108003/5f1044e79a86) (YouTube)
18. [ℚ is a topological field](https://invisiblenetwork.ai/learn/nptel/111108003/c9f4794b68e6) (YouTube)
19. [p-adic numbers](https://invisiblenetwork.ai/learn/nptel/111108003/1e42bff57116) (YouTube)
20. [p-adic integers](https://invisiblenetwork.ai/learn/nptel/111108003/4bd257ca4dc3) (YouTube)
21. [p-adic integers as a completion](https://invisiblenetwork.ai/learn/nptel/111108003/99b8ebba96fc) (YouTube)
22. [p-adic expansion of p-adic integers](https://invisiblenetwork.ai/learn/nptel/111108003/186e3e8fd0b9) (YouTube)
23. [Hensel's lemma](https://invisiblenetwork.ai/learn/nptel/111108003/29739168abd5) (YouTube)
24. [Dedekind Domains](https://invisiblenetwork.ai/learn/nptel/111108003/a1f17e2843fa) (YouTube)
25. [Unique Factorisation of Ideals in Dedekind Domains](https://invisiblenetwork.ai/learn/nptel/111108003/150003068d5f) (YouTube)
26. [Ring of Integers are Dedekind Domains](https://invisiblenetwork.ai/learn/nptel/111108003/55ae278713d0) (YouTube)
27. [Galois extensions](https://invisiblenetwork.ai/learn/nptel/111108003/a10259adb910) (YouTube)
28. [Decomposition Groups](https://invisiblenetwork.ai/learn/nptel/111108003/4d1fbcf644ea) (YouTube)
29. [Ramification indices and residue field degrees](https://invisiblenetwork.ai/learn/nptel/111108003/576e2472a7ec) (YouTube)
30. [efg theorem](https://invisiblenetwork.ai/learn/nptel/111108003/54371db47cf0) (YouTube)
31. [Inertia group](https://invisiblenetwork.ai/learn/nptel/111108003/322500d8fd4e) (YouTube)
32. [Class Groups](https://invisiblenetwork.ai/learn/nptel/111108003/60a8a1c787f9) (YouTube)
33. [Equivalent characterisation of invertible fractional ideals](https://invisiblenetwork.ai/learn/nptel/111108003/9bf66f864985) (YouTube)
34. [Basic PID](https://invisiblenetwork.ai/learn/nptel/111108003/5dd318a9eabd) (YouTube)
35. [Norm Estimates](https://invisiblenetwork.ai/learn/nptel/111108003/279a475b03b4) (YouTube)
36. [Finiteness of Class group I](https://invisiblenetwork.ai/learn/nptel/111108003/d12ba41f43c3) (YouTube)
37. [Finiteness of Class group II](https://invisiblenetwork.ai/learn/nptel/111108003/281a3c6c6f0e) (YouTube)