Algebraic Number Theory

Number theory is a study of Diophantine equations, in other words, polynomial equations with integer or rational coefficients for which we seek integer or rational solutions. For example, Pythagorean triplets x^2+y^2 = z^2. More generally, Fermat equation x^n+y^n = z^n. Another famous example is the Catalan equation x^n+1 = y^m etc. Although we look for integer solutions, it is often useful to extend our number systems and work in these extended number systems. For instance, x^2+y^2 factorises as (x+iy)(x-iy), where i is a square root of -1. Hence it makes more sense to work over Gaussian numbers Q(i). More generally, we can consider any polynomial f(x) with integer coefficients and obtain a finite extension of rational numbers by attaching roots of f(x) = 0. These are called number fields. In this Algebraic Number Theory course we study such number fields, and various objects attached to it. Some of these objects and invariants are

Ring of integers Units in the ring of integers

Discriminant

Different

Ideal class group

L-functions.

We will define these objects, study their properties, study algorithms to compute them, and prove several properties about them.