Counting monic irreducible polynomials $P$ in ${\mathbb{F}_q[X]}$ for which order of ${X\!\!\hspace{4.44443pt}(\@mod \; P)}$ is odd

Ballot, Christian

doi:10.5802/jtnb.572

Counting monic irreducible polynomials

P

𝔽_{q} [X]

for which order of

X (mod P)

is odd

Ballot, Christian ¹

¹ Département de Mathématiques, Université de Caen, Campus 2, 14032 Caen Cedex, France

Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 1, pp. 41-58.

Résumé
Abstract

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 $X (mod P)$ 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.

Hasse showed the existence and computed the Dirichlet density of the set of primes $p$ for which the order of $2 (mod p)$ 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 $X (mod P)$ 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.

MR Zbl

DOI : 10.5802/jtnb.572

Affiliations des auteurs :

Ballot, Christian ¹

¹ Département de Mathématiques, Université de Caen, Campus 2, 14032 Caen Cedex, France

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     title = {Counting monic irreducible polynomials $P$ in ${\mathbb{F}_q[X]}$ for which order of ${X\!\!\hspace{4.44443pt}(\@mod \; P)}$ is odd},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {41--58},
     publisher = {Universit\'e Bordeaux 1},
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Ballot, Christian. 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, Tome 19 (2007) no. 1, pp. 41-58. doi : 10.5802/jtnb.572. https://www.numdam.org/articles/10.5802/jtnb.572/

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