Algebraic Number theory January 2024

This is a course webpage of MT3224; 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-30%

2. End-Semester Exams-30%.

3. Class tests (4):-10%+10%+10%+10%.

Recommended Reading:

1. 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:

Tutors for the course:

1. Arup Mondal

2. Sutirth Dutta

Class test 1: January 19th. 

Class test 2: February 9 (syllabus-whatever taught till February 2).  

Prerequisites

Galois theory

Class timing and venue:

Monday:10-10.50 am 

Wednesday:9-9.50am

Friday: 10-10.50 am 

Venue

 LHC 205

Plan of the course:-

Week 1

1. Quadratic recipricity law. 

2.  Recall the definition of Euclidean domain, UFD, PID, Finite fields (Dummit and Foot and Pierre Samuel)

3. What is algebraic number theory?

Week 2

10/Ring of integers of number fields, 

 Integral elements. Definition, basic properties and examples of Integral elements.

Week 3

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

Discriminants:- Definition and basic properties of the Discriminants.

Week 4

Finding integral basis of O_K

Integral basis of cyclotomic fields

Week 5

Dedekind domains:- Definition, examples and counter examples of Dedekind domains.

Fractional ideals and it's unique factorization.

Week 8

Recall Chinese Remainder theorem. The formula n=\sum_i^r e_i f_i

Recall the wonderful formula n=\sum_i^r e_i f_i; Towards Dedekinds' theorem

Weak 9

How to find e_i, f_i, r in the wonderful formula as above.

Week 10

Mid-sem exam questions discussion. 

Kummer's theorem and how to find factorization. 

Week 11

Primes ramify if and only if it divide the discriminant.

Week 12

Fractional ideals and finiteness of class groups (Marcus's proof).

Minkowski's Lattice point theorem

Week 11

Non-teaching day.

How to calculate class groups.

Week 12:

Dirichlet's unit theorem.

Week 13

Decomposition groups and inertia groups.