A Multiquadratic Field Generalization of Artin's Conjecture

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 $-1$.