ANT16

Algebraic Number Theory MTH420

January - April 2016

Instructor : Dr. Anupam Singh

Schedule : Lectures - Monday, Wednesday 2-3 pm

Tutorial - Thursday 2-3 pm

Evaluation : Test I - 25 %

Test II (mid-sem exam) - 25%

Test III - mini projects - 25 %

Test IV - end-sem exam - 25 %

Prerequisites : preferably Field Theory

Goal of the course : Introduction to number theory, SAGE tutorials

Proposed course content :

Number Fields, Ring of Integers, Dedekind Domains, Solving Polynomials, Integral Bases.

Ideal factorization, Ideal Class Group, Finiteness of Class Group, Dirichlet's Unit Theorem, Solution of FLT in a special case.

Introduction to p-adic numbers.

Text books and references :

    1. Number Fields: Daniel A. Marcus, Universitext, Springer;

    2. Problems in Algebraic Number Theory: M. RamMurty and Jody Esmonde, GTM, Springer;

    3. Algebraic Number Theory, a computational approach : Stein

    4. Algebraic Number Theory and Fermat's last theorem: Stewart and Tall, A. K. Peters

    5. P-adic Numbers, p-adic Analysis, and Zeta functions : Koblitz, GTM Springer.

Topics for mini-project :

  1. PID but not ED (example of quadratic extension d= -19) - Papia

  2. Fermat’s Theorem for regular primes - Chris

  3. 2-Square, 4-Square theorem - Varun

  4. Pell’s equation - Chakraval method - Siddhartha

  5. Prime Number Theorem - Dileep

  6. Zeta function (Riemann, Dedekind) - Ajith

  7. Quadratic Reciprocity and higher reciprocity laws - Debarun

  8. Primality testing - Visakh

4 January 2016 - Motivation, Solving Equations, Diophantine equations

6 January 2016 - Translating the problem of solving integral equations over Z to an Algebra problem, Fermat's equation and Pell's equation

7 January 2016 - Pythagorean Triples, Solution of Fermat's equation for n=4.

Exercise set 1 : Exercises 1-9 from Marcus chapter 1

11 Jan 2016 - Number fields

13 Jan 2016 - Algebraic Integers, Ring of Integers in a number field O_K

14 Jan 2016 - Tutorial

Assignment I

18 Jan 2016 - Trace and Norm, Ring of integers in Quadratic number field

20 Jan 2016 - Discriminant

21 Jan 2016 - Tutorial

25 Jan 2016 - Additive structure of O_K, Integral Basis

27 Jan 2016 - Ring of integers in cyclotomic extensions

28 Jan 2016 - Tutorial

1 Feb 2016 - Dedekind domains, O_K is a DD

3 Feb 2016 - Ideal Class Group

4 Feb 2016 - Tutorial

Test I on 7 February at 10:30 AM

8 Feb 2016 - Unique factorization of ideals in to prime ideals for DD

10 Feb 2016 - Finiteness of ideal class group

11 Feb 2016 - Tutorial

Assignment 2

15 Feb 2016 - Computing Ideal class groups

17 Feb 2016 - Examples

18 Feb 2016 - Tutorial

Mid Sem Exam - 24 February at 10AM

29 February 2016 - Norm of an ideal, ramification index, residual degree

2 March 2016 - Splitting of primes (relation between ramification indices, residual degrees and degree of extension)

3 March 2016 - Tutorial - splitting of primes in quadratic extensions vs Legendre symbols

I am away for a conference so no lecture during the week 7-11 March (classes will be substituted later)

14 March 2016 - Geometric idea, lattices in R^n

16 March 2016 - Minkowaski's Theorem

17 March 2016 - Minkowaski bound on Ideal Class Group

20 March 2016 - SAGE tutorial (computing factorization of ideals and finding ideal class group)

23 March 2016 - Drichilet's unit theorem

24 March 2016 - Holiday

28 March 2016 -2-Square, 4-Square theorem - Varun

30 March 2016 - Pell’s equation - Chakraval method - Siddhartha

31 March 2016 - Quadratic Reciprocity and higher reciprocity laws - Debarun

4 April 2016 - PID but not ED (example of quadratic extension d= -19) - Papia

6 April 2016 - Fermat’s Theorem for regular primes - Chris

7 April 2016 - Primality testing - Visakh

8 April 2016 - Zeta function (Riemann, Dedekind) - Ajith

Prime Number Theorem - Dileep

11 April 2016 - p-adic numbers, Hasse-Minkowaski theorem

13 April 2016 - Summary