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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.

MSC:

11R44 Distribution of prime ideals
11R11 Quadratic extensions

Citations:

Zbl 0221.10048
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References:

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