A multiquadratic field generalization of Artin's conjecture

Abstract

We prove (under the assumption of the generalized Riemann hypothesis) that a totally real multiquadratic number field K has a positive density of primes p in Z for which the image of OK× in (OK/pOK)× has minimal index (p−1)/2 if and only if K contains a unit of norm −1. An explicit formula for this density is provided. We also discuss an application to ray class fields of conductor pOK.