Constructing Extensions of Number Fields

By Jonah Leshin

November 30, 2011

Abstract

A number field is a finite extension of the rational numbers. While we understand many properties of number fields in the abstract (e.g. the discriminant, the ideal class group), there are many open questions concerning the existence of number fields with various properties. The tools used to solve these problems include class field theory, modular forms, and arithmetic geometry. In this talk I will discuss how some of these tools have been used in this context. I will not assume any background beyond Galois theory and a preference for pizza over samosas.