A multiquadratic field generalization of Artin's conjecture
type: Item
instance of: scholarly article
title: A multiquadratic field generalization of Artin's conjecture
zbMATH Open document ID: 1409.11096
author: M. E. Stadnik
published in: Journal of Number Theory
publication date: 2016-09-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1202.3475
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11R45
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 6626756
zbMATH Keywords: Artin's conjecture
zbMATH Keywords: Frobenius elements
zbMATH Keywords: multiquadratic fields
Wikidata QID: Q123245409
MaRDI profile type: Publication
OpenAlex ID: W2229383465
arXiv ID: 1202.3475
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DOI: 10.1016/J.JNT.2016.06.012
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review text: Among the conjectures due to Artin there is one that states that for any nonsquare integer \(a \neq -1\) the set of primes for which \(a\) is a primitive root modulo \(p\) has positive density. The present article deals with the following more general question: For which number fields \(K\) is the index of the unit group in \(({\mathcal O}_K/p{\mathcal O}_K)^\times\) equal to \(\frac{p-1}2\) for infinitely many \(p\)? It is easy to see that the index is at least \(p-1\) if each unit has norm \(+1\).NEWLINENEWLINEThe main result proved in this article is the following theorem: Let \(K\) be a totally real multiquadratic number field with unit grooup \({\mathcal O}_K^\times\), and assume that \(K\) contains a unit with norm \(-1\). Then the generalized Riemann Hypothesis implies that the primes that split in \(K\) and for which the image of \({\mathcal O}_K^\times\) in \(({\mathcal O}_K/p{\mathcal O}_K)^\times\) has index \((p-1)/2\) has a positive density, which can be given explicitly.
description: scientific article; zbMATH DE number 6626756