Dirichlet's Lemma in Number Fields

TL;DR

This paper introduces the separant class group to measure the failure of Dirichlet's Lemma in general number fields and explores its implications for quadratic extension genus theory.

Contribution

It defines the separant class group and demonstrates its role in extending explicit genus theory to number fields with trivial separant class groups.

Findings

Separable class group measures Dirichlet's Lemma failure

Fields with trivial separant class groups allow explicit genus theory

Application to quadratic extensions in number fields

Abstract

Dirichlet's Lemma states that every primitive quadratic Dirichlet character $\chi$ can be written in the form $\chi(n) = (\frac{\Delta}n)$ for a suitable quadratic discriminant $\Delta$. 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.