Artin's primitive root conjecture for quadratic fields

Hans Roskam

doi:10.5802/jtnb.360

Hans Roskam

Journal de Théorie des Nombres de Bordeaux, Tome 14 (2002) no. 1, pp. 287-324.

Résumé
Abstract

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.

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.

MR 1926004 | Zbl 1026.11086

DOI : https://doi.org/10.5802/jtnb.360

@article{JTNB_2002__14_1_287_0,
     author = {Roskam, Hans},
     title = {Artin's primitive root conjecture for quadratic fields},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {287--324},
     publisher = {Universit\'e Bordeaux I},
     volume = {14},
     number = {1},
     year = {2002},
     doi = {10.5802/jtnb.360},
     zbl = {1026.11086},
     mrnumber = {1926004},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.360/}
}

Hans Roskam. Artin's primitive root conjecture for quadratic fields. Journal de Théorie des Nombres de Bordeaux, Tome 14 (2002) no. 1, pp. 287-324. doi : 10.5802/jtnb.360. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.360/

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