Instructor: Dr Anupam Singh
Audience: 3rd year BS-MS, MSc students at IISER Pune
Schedule: Tuesday, Wednesday 10-10:550 AM (LHC305)
Tutorial: Friday 11-11:55 AM (LHC305)
Evaluation: Test I - 20%, mid-sem - 30%, Test II - 20%, end-sem - 30%
Prerequisite: Rings and Fields
Goal of the course: Solving equations is a fundamental problem in Mathematics. Solving polynomial equations over integers or rationals can be translated to algebra problems, more specifically, ring theory questions. Several techniques have been developed, and the course aims to provide a flavour of them. The solution of Fermat's last theorem a couple of decades ago was one of the major milestones in the history of mathematics, which can be appreciated through this course. Students with a strong background in algebra are ideally suited for this course.
Proposed Content: Algebraic numbers and algebraic integers: definitions and basic properties, Integral extensions, Dedekind domains, prime ideals, ideal factorization.
The law of quadratic reciprocity uses number fields, Minkowaski's Theorem, finiteness of class groups, Dirichlet’s Unit Theorem, and class number formula.
Solution of FLT in a special case. Introduction to p-adic numbers.
Introduction to SAGEMATH for computation.
Textbooks and reference material :
Number Fields: Daniel A. Marcus
Problems in Algebraic Number Theory: M. RamMurty and Jody Esmonde
Algebraic Number Theory, a computational approach : Stein
Algebraic Number Theory and Fermat's last theorem: Stewart and Tall
P-adic Numbers, p-adic Analysis, and Zeta functions : Koblitz
Project Topics:
Fermat’s Theorem for regular primes – Gaurav Verma
2-Square, 4-Square theorem - Goutham Krishnakumar
Pell’s equation - Chakraval method — Aniket Datta
Prime Number Theorem - Anshuka Laddikam
Riemann Zeta function - Nipurn Shakya
Quadratic Reciprocity and Higher Reciprocity laws – Drishti Sunder Phukon
Transcendental numbers –Abhijeet Mohanty
Weekly topics covered :
03/01/25: Course information, Motivation, Diophantine equations, Fermat's, etc
07/01/25: Algebraic integers, examples
08/01/25: Ring of integers in quadratic, Trace, and norm
10/01/25: Tutorial
14/01/25: Holiday
15/01/25: Discriminant
17/01/25: Tutorial
21/01/25: Integral basis
21/01/25 (extra class 1 hour): Ring of integers of cyclotomic fields
22/01/25: Dedekind Domain, O_K is DD
24/01/25: Ideal class group for a Dedekind Domain
25/01/25 (extra class 2 hours): Tutorial + Computing with number fields in SageMath
28-29/01/25 and 4-5/02/25: no class
31/01/25: Ideals are a product of prime ideals in DD.
07/02/25: Tutorial
08/02/25: Test (2 PM)
11/02/25: Ramification index, residue field, and inertial degree
12/02/25: Splitting of primes
14/02/25: tutorials
18/02/25: Example of splitting of prime in quadratic and Legendre symbol
Mid-Sem exam - 22 Feb 25 at 3 PM.
04/03/25: Prime splitting and Galois extension, ramified primes and discriminant
05/03/25: Finiteness of Class Group
07/03/25: Tutorial
11/03/25: No class
12/03/25: Geometric ideas, lattice, lattice of O_K
14/03/25: Holiday
18/03/25: Minkowaski's Lemma and Minkowski bound (statement)
19/03/25: Computation of class group - examples
Project presentations 3-4 and 4-5
21/03/25: Project presentation
25/03/25: Minkowski bound for ideals in number fields
26/03/25: Some applications
Project presentations 3-4 and 4-5
28/03/25: Project presentation
01/04/25: Project presentation
02/04/25: Project presentation
04/04/25: Project presentation
08/04/25: Logarithmic space
09/04/25: Dirichlet's Unit Theorem
11/04/25: SageMath - computing Minkowski bound, Class number, units etc
15/04/25: p-adic numbers
16/04/25: Class number formula, Gauss' class number problems
End Sem Exam: 30/04/25