Dirichlet's Lemma in Number Fields
Franz Lemmermeyer
TL;DR
The paper introduces the separant class group ${\operatorname{SCl}}(F)$ to quantify the failure of Dirichlet's Lemma in general number fields and proves ${\operatorname{SCl}}(F) \cong {\operatorname{Cl}}_F\{4\}/{\operatorname{Cl}}_F\{4\}^2$, with size $2^{\rho^+ + s}$ where $\rho^+$ is the 2-rank of the strict class group and $s$ is the number of complex embeddings. It shows ${\operatorname{SCl}}(F)=1$ precisely when $F$ is totally real with odd strict class number, in which case separants factor uniquely into prime separants and genus theory becomes explicit, mirroring the rational case. The work establishes a bridge between Kronecker and Dirichlet characters via ${\operatorname{SCl}}(F)$, leveraging ray class groups and exact sequences to provide concrete descriptions of genus fields, principal genus, and ramification patterns for quadratic extensions over $F$. Overall, the results extend genus theory, separant concepts, and Hilbert-type reciprocity laws to a broad class of number fields, enabling explicit arithmetic of quadratic extensions.
Abstract
Dirichlet's Lemma states that every primitive quadratic Dirichlet character $χ$ can be written in the form $χ(n) = (\fracΔn)$ for a suitable quadratic discriminant $Δ$. 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.