Counting monic irreducible polynomials P in 𝔽 q [X] for which order of X(modP) is odd
Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 41-58.

Hasse showed the existence and computed the Dirichlet density of the set of primes p for which the order of 2(modp) is odd; it is 7/24. Here we mimic successfully Hasse’s method to compute the density δ q of monic irreducibles P in 𝔽 q [X] for which the order of X(modP) is odd. But on the way, we are also led to a new and elementary proof of these densities. More observations are made, and averages are considered, in particular, an average of the δ p ’s as p varies through all rational primes.

Hasse démontra que les nombres premiers p pour lesquels l’ordre de 2 modulo p est impair ont une densité de Dirichlet égale à 7/24-ième. Dans cet article, nous parvenons à imiter la méthode de Hasse afin d’obtenir la densité de Dirichlet δ q de l’ensemble des polynômes irréductibles et unitaires P de l’anneau 𝔽 q [X] pour lesquels l’ordre de X(modP) est impair. Puis nous présentons une seconde preuve, nouvelle, élémentaire et effective de ces densités. D’autres observations sont faites et des moyennes de densités sont calculées, notamment la moyenne des δ p lorsque p parcourt l’ensemble des nombres premiers.

Received:
Published online:
DOI: 10.5802/jtnb.572
Christian Ballot 1

1 Département de Mathématiques, Université de Caen, Campus 2, 14032 Caen Cedex, France
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Christian Ballot. Counting monic irreducible polynomials $P$ in ${\mathbb{F}_q[X]}$ for which order of ${X\!\!\hspace{4.44443pt}(\@mod \; P)}$ is odd. Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 41-58. doi : 10.5802/jtnb.572. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.572/

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