Artin's primitive root conjecture for quadratic fields
type: Item
instance of: scholarly article
title: Artin's primitive root conjecture for quadratic fields
zbMATH Open document ID: 1026.11086
author: Hans Roskam
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2003-06-30T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/232558
review text: Artin's conjecture on primitive roots says that the set of primes for which a given integer \(a\) is a primitive root has a natural density, which is positive except when \(a = \pm 1\). There are two ways in which this conjecture can be generalized to arbitrary number fields: the obvious one, where the set of primes is replaced by the set of prime ideals, was proved under the assumption of GRH by \textit{G. Cooke} and \textit{P. Weinberger} [Commun. Algebra 3, 481-524 (1975; Zbl 0315.12001)]. The conjecture studied in this article is the following: let \(O\) denote the ring of integers of the number field \(K\), and consider the rational primes for which the order of a given \(\alpha \in K^\times\) inside the group \(C_p = (O/pO)^\times\) is equal to the exponent of \(C_p\); then this set has a natural density inside the set of all rational primes. The author proves this conjecture for quadratic number fields again assuming the truth of GRH and explains why, at least at present, this proof cannot be generalized to fields of degree \(> 2\).
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11R44
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 1943341
zbMATH Keywords: Artin's conjecture
zbMATH Keywords: primitive roots
zbMATH Keywords: quadratic fields
zbMATH Keywords: generalized Riemann hypothesis
MaRDI profile type: Publication
Wikidata QID: Q123244447
OpenAlex ID: W2317742326
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DOI: 10.5802/JTNB.360
description: scientific article; zbMATH DE number 1943341