Dirichlet's Lemma in Number Fields

Abstract

Dirichlet's Lemma states that every primitive quadratic Dirichlet character $\ddot{I}‡$ can be written in the form $\ddot{I}‡(n)=(\frac{\hat{I}}{”}n)$ for a suitable quadratic discriminant $\hat{I}”$. In this article we define a group, the separant class group, that measures the extent to which Dirichlet's Lemma fails in general number fields $F$. As an application we will show that over fields with trivial separant class groups, genus theory of quadratic extensions can be made as explicit as over the rationals.

submitted to Acta Arithmetica