A quadratic analogue of Artin's conjecture on primitive roots.
type: Item
instance of: scholarly article
title: A quadratic analogue of Artin's conjecture on primitive roots.
zbMATH Open document ID: 1049.11125
author: Hans Roskam
published in: Journal of Number Theory
publication date: 2000-12-12T00:00:00Z
review text: Let \(\epsilon\) be a fundamental unit in a real quadratic field \(k\), and let \(S\) be the set of rational primes \(p\) for which \(\varepsilon\) has maximal order modulo \(p\). Under the assumption of the generalized Riemann hypothesis, we show that \(S\) has a density \(\delta (S)=c\cdot A\) in the set of all rational primes, where \(A\) is Artin's constant and \(c\) is a positive rational number. In the proof the author uses techniques of \textit{C. Hooley} [J. Reine Angew. Math. 225, 209--220 (1967; Zbl 0221.10048)] and results of the splitting behaviour of primes in subfields of the Kummer extension \(\mathbb Q(\zeta_l,{\root l\of\varepsilon})\). In the Erratum the statements of Lemma 6, Lemma 7c, Theorem 2a, and Theorem 2b are corrected.
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11R44
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 1420265
zbMATH Keywords: Artin's primitive root conjecture
MaRDI profile type: Publication
OpenAlex ID: W2021439610
cites work: Q5522902
cites work: On Artin's conjecture and Euclid's algorithm in global fields
cites work: On Artin's conjecture.
cites work: On the distribution of units modulo prime ideals in real quadratic fields
cites work: Q5287502
cites work: On Artin's conjecture
cites work: Q4773795
Wikidata QID: Q123102815
DOI: 10.1006/JNTH.1999.2470
description: scientific article; zbMATH DE number 1420265