Dirichlet’s Lemma in number fields

Dirichlet’s Lemma states that every primitive quadratic Dirichlet character χ can be written in the form χ(n)=(Δn) for a suitable quadratic discriminant Δ. In this article we define a group, the separant class group SCl(F), that measures the extent to which Dirichlet’s Lemma fails in general number fields F.