A multiquadratic field generalization of Artin's conjecture

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• 作者：M.E. Stadnik ^{mstadnik@bsc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Artin's conjecture ; Frobenius elements ; Multiquadratic fields
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：170
• 期：Complete
• 页码：75-102
• 全文大小：528 K

文摘

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$\mathbb{Z}$ for which the image of ${\mathcal{O}}_K^{\times }$ in (O_K/pO_K)^×${({\mathcal{O}}_K/p{\mathcal{O}}_K)}^{\times }$ has minimal index (p−1)/2$(p-1)/2$ 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 pO_K$p{\mathcal{O}}_K$.