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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 onnumber fields. We prove that the natural density of the set of integersrepresented by a norm form is zero, while the relative natural density of theset of prime numbers represented by a norm form exists and is positive. Theseresults require tools from class field theory, including the Artin-Chebotarevdensity theorem and the Hilbert class field. We introduce these tools as weneed them in the course of the main arguments. This article is expository innature and assumes only a first course in algebraic number theory.

Norm forms represent few integers but relatively many primes