A Multiquadratic Field Generalization of Artin's Conjecture

TL;DR

This paper proves, assuming the generalized Riemann hypothesis, that totally real multiquadratic fields contain a positive density of primes with a specific unit group property if and only if they have a unit of norm -1.

Contribution

It generalizes Artin's conjecture to multiquadratic fields, establishing a criterion for the density of primes based on units of norm -1.

Findings

Positive density of primes with minimal index in unit groups under GRH

Characterization of multiquadratic fields containing units of norm -1

Conditional proof extending Artin's conjecture to new class of fields

Abstract

We prove that (under the assumption of the generalized Riemann hypothesis) a totally real multiquadratic number field $K$ has a positive density of primes $p \in \mathbb{Z}$ for which the image of the unit group $(\mathcal{O}_K)^{\times})$ in $(\mathcal{O}_K/p\mathcal{O}_K)^{\times})$ has minimal index $(p-1)/2$ if and only if $K$ contains a unit of norm $-1$.