The density of primes represented by norm forms

Abstract

A norm form is a multivariate homogeneous polynomial with integral coefficients, such as x^2 + y^2 or x^2 + xy − y^2, arising from the norm function of a number field. In this thesis, we present a formula for the density of prime numbers represented by the absolute value of a norm form. We use techniques from class field theory to show that the density formula only depends on the Galois group of the Galois closure of the Hilbert class field of the number field. We find that when the original field is Galois, the Hilbert class field is also Galois, and describe how this simplifies the density formula. We also present analogous results for primes represented by norm forms, without the absolute value. Finally, we discuss the progress made on determining the Galois group of the Galois closure of the Hilbert class field in the case that the original field has prime degree and class number 2.