A quadratic analogue of Artin’s conjecture on primitive roots. (English) Zbl 1049.11125

J. Number Theory 81, No. 1, 93-109 (2000); Erratum ibid. 85, No. 1, 108 (2000).
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 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.


11R44 Distribution of prime ideals
11R11 Quadratic extensions


Zbl 0221.10048
Full Text: DOI Link


[1] Artin, E, ()
[2] Lenstra, H.W, On Artin’s conjecture and Euclid’s algorithm in global fields, Inv. math., 42, 201-224, (1977) · Zbl 0362.12012
[3] Hooley, C, On Artin’s conjecture, J. reine angew. math., 225, 209-220, (1967) · Zbl 0221.10048
[4] Ishikawa, M; Kitaoka, Y, On the distribution of units modulo prime ideals in real quadratic fields, J. reine angew. math., 494, 65-72, (1998) · Zbl 0883.11046
[5] S. Lang, Algebra, third edition, Addison-Wesley, 1993.
[6] Ram Murty, M, On Artin’s conjecture, J. number theory, 16, 147-168, (1983) · Zbl 0526.12010
[7] Weinberger, P.J, On Euclidean rings of algebraic integers, Analytic number theory, proc. sympos. pure math., XXIV, 321-332, (1973)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.