Artin's primitive root conjecture for quadratic fields

Hans Roskam

Journal de théorie des nombres de Bordeaux (2002)

  • Volume: 14, Issue: 1, page 287-324
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Roskam, Hans. "Artin's primitive root conjecture for quadratic fields." Journal de théorie des nombres de Bordeaux 14.1 (2002): 287-324. <http://eudml.org/doc/248915>.

@article{Roskam2002,
abstract = {Fix an element $\alpha $ in a quadratic field $K$. Define $S$ as the set of rational primes $p$, for which $\alpha $ 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.},
author = {Roskam, Hans},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Artin's conjecture; primitive roots; quadratic fields; generalized Riemann hypothesis},
language = {eng},
number = {1},
pages = {287-324},
publisher = {Université Bordeaux I},
title = {Artin's primitive root conjecture for quadratic fields},
url = {http://eudml.org/doc/248915},
volume = {14},
year = {2002},
}

TY - JOUR
AU - Roskam, Hans
TI - Artin's primitive root conjecture for quadratic fields
JO - Journal de théorie des nombres de Bordeaux
PY - 2002
PB - Université Bordeaux I
VL - 14
IS - 1
SP - 287
EP - 324
AB - Fix an element $\alpha $ in a quadratic field $K$. Define $S$ as the set of rational primes $p$, for which $\alpha $ 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.
LA - eng
KW - Artin's conjecture; primitive roots; quadratic fields; generalized Riemann hypothesis
UR - http://eudml.org/doc/248915
ER -

References

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  2. [2] H. Bilharz, Primdivisoren mit vorgegebener Primitivwurzel. Math. Ann.114 (1937), 476-492. Zbl0016.34301MR1513151JFM63.0099.01
  3. [3] G. Cooke, P.J. Weinberger, On the construction of division chains in algebraic number fields, with applications to SL2. Commun. Algebra3 (1975), 481-524. Zbl0315.12001MR387251
  4. [4] H.W. Lenstra, JR, On Artin's conjecture and Euclid's algorithm in global fields. Inv. Math.42 (1977), 201-224. Zbl0362.12012MR480413
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  6. [6] P. Moree, Approximation of singular series and automata. Manuscripta Math.101 (2000), 385-399. Zbl1007.11084MR1751040
  7. [7] M. Ram Murty, On Artin's Conjecture. J. Number Theory16 (1983), 147-168. Zbl0526.12010MR698163
  8. [8] H. Roskam, A quadratic analogue of Artin's conjecture on primitive roots. J. Number Theory81 (2000), 93-109. Errata in J. Number Theory85 (2000), 108. Zbl1049.11125MR1743503
  9. [9] H. ROSKAM Prime divisorsof linear recurrences and Artin's primitive root conjecture for number fields. J. Théor. Nombres Bordeaux13 (2001), 303-314. Zbl1044.11005MR1838089
  10. [10] J.-P. Serre, Quelques applications du théorème de densité de Chebotarev. Inst. Hautes Études Sci. Publ. Math.54 (1981), 323-401. Zbl0496.12011MR644559
  11. [11] J.-P. Serre, Local Fields (2nd corrected printing). Springer-Verlag, New York, 1995. MR554237
  12. [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. Zbl0287.12012MR337902

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