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 :
Number Fields: Daniel A. Marcus, Universitext, Springer;
Problems in Algebraic Number Theory: M. RamMurty and Jody Esmonde, GTM, Springer;
Algebraic Number Theory and Fermat's last theorem: Stewart and Tall, A. K. Peters
P-adic Numbers, p-adic Analysis, and Zeta functions : Koblitz, GTM Springer.
Topics for mini-project :
PID but not ED (example of quadratic extension d= -19) - Papia
Fermat’s Theorem for regular primes - Chris
2-Square, 4-Square theorem - Varun
Pell’s equation - Chakraval method - Siddhartha
Prime Number Theorem - Dileep
Zeta function (Riemann, Dedekind) - Ajith
Quadratic Reciprocity and higher reciprocity laws - Debarun
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