## 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

### Keywords:

Artin’s primitive root conjecture

Zbl 0221.10048
Full Text:

### References:

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