Number Theory and Algebra

Project Description

The goal of this project is to give a gentle introduction to Modern Number theory. We will cover basic topics like divisibility, modular arithmetic, Euler’s theorem, fundamental theorem of arithmetic, quadratic residues and Gauss’s famous Quadratic reciprocity law. Along the way, we will introduce several key algebraic objects like groups, rings and fields. If time permits, we will talk about some well known problems in the field like Dirichlet’s theorem for primes in Arithmetic progressions and the Riemann hypothesis.

Mentors

• Ajay Prajapati
• Hargun Preet Singh Bhatia

Learning Roadmap

1st & 2nd week: We covered the very basics of linear algebra, focussing mainly on the concept of a vector space, basis, dimension, etc.

3rd week: We covered basic set theoretic concepts like equivalence relations and described the set of integers mod n.

4th-8th week: We covered the fundamentals of groups, homomorphisms, and group actions.

9th-11th week: We covered the fundamentals of ring theory. Main highlight was the proof of Fermat’s theorem on the sum of 2 squares using properties of the ring of Gaussian integers. We then also ventured onto polynomial rings and covered some examples of finite fields of prime power order.

Then we gave a final exercise on Quadratic integer rings.

Resources

Prof. Abhijit Pal’s notes on Linear Algebra, Topics in Algebra by I.N. Herstein, Abstract Algebra by Dummit and Foote.