Artin's primitive root conjecture for quadratic fields
Journal de Théorie des Nombres de Bordeaux, Volume 14 (2002) no. 1, pp. 287-324.

Fix an element α in a quadratic field K. Define S as the set of rational primes p, for which α has maximal order modulo p. Under the assumption of the generalized Riemann hypothesis, we show that S has a density. Moreover, we give necessary and sufficient conditions for the density of S to be positive.

Soit α fixé dans un corps quadratrique K. On note S l’ensemble des nombres premiers p pour lesquels α admet un ordre maximal modulo p. Sous G.R.H., on montre que S a une densité. Nous donnons aussi des conditions nécessaires et suffisantes pour que cette densité soit strictement positive.

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     title = {Artin's primitive root conjecture for quadratic fields},
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     publisher = {Universit\'e Bordeaux I},
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Hans Roskam. Artin's primitive root conjecture for quadratic fields. Journal de Théorie des Nombres de Bordeaux, Volume 14 (2002) no. 1, pp. 287-324. doi : 10.5802/jtnb.360. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.360/

[1] E. Artin, The collected papers of Emil Artin, (eds S. Lang, J. Tate). Addison-Wesley, 1965. | MR: 176888 | Zbl: 0146.00101

[2] H. Bilharz, Primdivisoren mit vorgegebener Primitivwurzel. Math. Ann. 114 (1937), 476-492. | JFM: 63.0099.01 | MR: 1513151 | Zbl: 0016.34301

[3] G. Cooke, P.J. Weinberger, On the construction of division chains in algebraic number fields, with applications to SL2. Commun. Algebra 3 (1975), 481-524. | MR: 387251 | Zbl: 0315.12001

[4] H.W. Lenstra, JR, On Artin's conjecture and Euclid's algorithm in global fields. Inv. Math. 42 (1977), 201-224. | MR: 480413 | Zbl: 0362.12012

[5] C. Hooley, On Artin's conjecture. J. Reine Angew. Math. 225 (1967), 209-220. | MR: 207630 | Zbl: 0221.10048

[6] P. Moree, Approximation of singular series and automata. Manuscripta Math. 101 (2000), 385-399. | MR: 1751040 | Zbl: 1007.11084

[7] M. Ram Murty, On Artin's Conjecture. J. Number Theory 16 (1983), 147-168. | MR: 698163 | Zbl: 0526.12010

[8] H. Roskam, A quadratic analogue of Artin's conjecture on primitive roots. J. Number Theory 81 (2000), 93-109. Errata in J. Number Theory 85 (2000), 108. | MR: 1743503 | Zbl: 1049.11125

[9] H. Roskam Prime Divisors of linear recurrences and Artin's primitive root conjecture for number fields. J. Théor. Nombres Bordeaux 13 (2001), 303-314. | Numdam | MR: 1838089 | Zbl: 1044.11005

[10] J.-P. Serre, Quelques applications du théorème de densité de Chebotarev. Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323-401. | Numdam | MR: 644559 | Zbl: 0496.12011

[11] J.-P. Serre, Local Fields (2nd corrected printing). Springer-Verlag, New York, 1995. | MR: 554237

[12] P.J. Weinberger, On euclidean rings of algebraic integers. Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), 321-332. Amer. Math. Soc., Providence, R. I., 1973. | MR: 337902 | Zbl: 0287.12012

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