The density of primes represented by norm forms
Summary
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.