This is the course homepage of MTH 420 for Spring 2014. In this website, you will find the relevant course materials.
1. Lecture 1 (06/01/2014):-Introduction and motivation of algebraic number theory. In this lecture, I discussed the basic problems that we are going to study in this course.
2 . Lecture 2 (08/01/2014):-Integral elements. Definition, basic properties and examples of Integral elements.
3. Lecture 3 (09/01/2014):-Dedekind domains:- Definition, examples and counter examples of Dedekind domains.
4. Lecture 4 (13/01/2014):-Discriminants:- Definition and basic properties of the Discriminants.
5. Lecture 5 (15/01/2014):-Finite generation of ring of integers of number fields.
6. Lecture 6 (16/01/2014):-Fractional ideals. I gave assignment 1.
7. Lecture 7 (20/01/2014):Class group.
8. Lecture 8 (22/01/2014):- Chinese Remainder theorem.
9. Lecture 9 (23/01/2014):- Tutorial. I gave some problems on the Discriminants.
10. Lecture 10 (27/01/2014):- Some applications of Chinese Remainder theorem. We gave some interesting and important applications of the Chinese Remainder theorem including the fact that every ideal in O_K is generated by less than or equal to 2
elements.
11. Lecture 11 (29/01/2014):- Ramification index and inertia degree. We defined ramification index and inertia degrees. We prove an important theorem relating these quantities with extension degree
12. Lecture 12 (30/01/2014):- Unit Test 1.
13. Lecture 13 (03/02/2014):- Ramification indices and inertia degrees.
14. Lecture 14(05/02/2014):- How to find ramification indices and inertia degrees.
15. Lecture 15 (06/02/2014):- Tutorial.
16. Lecture 16 (10/02/2014):-Relation between inertia degrees, ramification indices and extension degrees.
17. Lecture 17 (12/02/2014):- Primes ramify if and only if it divide the discriminant.
18. Lecture 18 (13/02/2014):- Tutorial.
19. Lecture 19(17/02/2014) :-Eisenstein extensions. Properties of them including the index of monic generators.
20. Lecture 20 (19/02/2014):-Decomposition groups and inertia groups.
21. Lecture 21 (20/02/2014):-Tutorial.
22. Mid-Semester Examination (24/02/2014).
23. Lecture 22 (03/03/2014):-Lattices.
24. Lecture 23 (05/03/2014):- Minkowski's lattice point theorem.
25. Lecture 24 (06/03/2014):- Applications of Minkowski's lattice point theorem.
26. Lecture 25 (10/03/2014):-Finiteness of the class group.
27. Lectuure 26 (12/03/2014):- Computing the class group.
28. Presentation by students (13/03/2014):- 1. Ideal factorization in a Dedekind domian.
2. Kummer's factorization theorem.
29. Lecture 28 (17/03/2014). Dirichlet 's Unit theorem
30. Lecture 29 (19/03/2014). Applications of Unit theorem.
31. Presentation by students (20/03/2014) 1. Ramification indices and inertia degrees.
2. Ramifications in Eisenstein extensions.
32. Lecture 30 (26/03/2014). Discrete valuation rings.
33. Unit test (27/03/2014).
34. Lecture 31 (02/04/2014). Properties of Discrete valuation rings.
35. Presentation by students (03/04/2014) 1. Ramified primes and discriminants.
2. Finite generation of ring of integers.
36. Lecture 32 (07/04/2014). p-adic fields.
37. Lecture 33 (09/04/2014). Hensel's Lemma.
38. Lecture 34 (10/04/2014). Class number formula, finiteness of class group (student presentation).
39. Lecture 35 (14/04/2014). Structure theorem of p-adic fields, adeles and ideles.
40. Lecture 36 (16/04/2014). Tutorial/statement of Kronecker-Weber theorem.
17/04/2014. End of Semester.