MTH 420 (Algebraic number theory)

This is a course webpage of MTH 420; an undergraduate course on algebraic number theory. Please read the course template for text-books and other important information about the course. This is a 4 credit course.

Grading :

1. Mid-Semester Exams-35%
2. End-Semester Exams-35%.

3. Class tests (2):-15%+15%.

Please note that there will not be any repeat class tests.

Recommended Reading:

1. Number fields-Marcus
2. Algebraic number theory (Neukirch).
3. Algebraic theory of numbers-Pierre Samuel
4. Problems in algebraic number theory: Esmonde and Ram Murty

Announcements:

1. Class test I on 17/1/20.

2. Class test II on 7/2/20.

3. Mid-sem on 22/2/2020.

4. Class room shifted to Google classroom. Please check details in the email.

Prerequisites

MTH 410 (Galois theory)

Class timing:

Mon-5-5.55 pm

Tue-5-5.55 pm

Friday-12-12.55 pm

Class room:-

304

Plan of the course:-

Week 1

03/01/2020-What is algebraic number theory?

Recall the definition of Euclidean domain, UFD, PID (Dummit and Foot)

Week 2

6/1/20: Finite fields and it's automorphisms, Frobenius

7/1/20: Legendre symbols and law of quadratic reciprocity. Ref: Esmonde and Ram Murty

10/1/20: Tutorial. Assignment 1 uploaded (please see at the end of the page).

Week 3

13/1/20: Integral elements. Definition, basic properties and examples of Integral elements.

14/1/20: Ring of integers of number fields, non-degenerate bilinear forms. O_K is free!

17/1/20: Class test I; Assignment 2 uploaded.

Week 5

20/1/20:Trace induces a non-degnerate bilinear form, O_K is free (again), other properties of O_K.

21/1/20: Discriminants:- Definition and basic properties of the Discriminants.

24/1/20: Tutorial: Assignment 3 uploaded.

Week 6

27/1/20: Finding integral basis of O_K

28/1/20: Dedekind domains:- Definition, examples and counter examples of Dedekind domains.

31/1/20:Tutorial; Assignment 4 uploaded.

Week 7

3/2/20: Fractional ideals and it's unique factorization.

4/2/20: Recall chinese Remainder theorem. The formula n=\sum_i^r e_i f_i

7/2/20: Class test II. Assignment 5 uploaded.

Week 8

10/2/20: Recall the wonderful formula n=\sum_i^r e_i f_i; Towards Dedekinds' theorem

11/2/20: How to find e_i, f_i, r in the wonderful formula as above.

14/2/20: Tutorial

Weak M

17/2/20: Recap/ tutorial

18/2/20: Recap

22/2/20: Mid-sem

Week 9

28/2/20: Mid-sem exam questions discussion.

Week 10

02/03/20: Fractional ideals and finiteness of class groups (Marcus's proof).

03/03/20: Minkowski's Lattice point theorem

06/03/20:Assignment 6 uploaded, Tutorial

Week 11

09/03/20: Non-teaching day.

10/03/20: Holi (Holiday)

13/03/20: How to calculate class groups.

Week 12

Classes suspended due to Covid-19 pandemic.

Week 13

Classes suspended due to Covid-19 pandemic.

Class room shifted to Google classroom. Please check details in the email.

Week 14:

12/05/2020 at 5 pm :Dirichlet's unit theorem.

Week 15

19/05/2020 at 5 pm: Decomposition groups and inertia groups.

Week 16

26/05/20:Introduction to \Q_p and \Z_p

Week 15

02/06/20: Infinitude of primes, Riemann and Dedekind Zeta functions and class number formula

Week 16

09/06/20: Revision

Week 17:

Assignments:-