  Norm forms, examples of which include $x^2 + y^2$, $x^2 + x y - 57 y^2$, and
$x^3 + 2 y^3 + 4 z^3 - 6 x y z$, are integral forms arising from norms on
number fields. We prove that the natural density of the set of integers
represented by a norm form is zero, while the relative natural density of the
set of prime numbers represented by a norm form exists and is positive. These
results require tools from class field theory, including the Artin-Chebotarev
density theorem and the Hilbert class field. We introduce these tools as we
need them in the course of the main arguments. This article is expository in
nature and assumes only a first course in algebraic number theory.

Number Theory