Counting monic irreducible polynomials $P$ in ${𝔽}_q\left[X\right]$ for which order of $X(modP)$ is odd

Ballot, Christian

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 ${\delta }_q$ of monic irreducibles $P$ in ${𝔽}_q\left[X\right]$ 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 ${\delta }_p$’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 ${\delta }_q$ de l’ensemble des polynômes irréductibles et unitaires $P$ de l’anneau ${𝔽}_q\left[X\right]$ 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 ${\delta }_p$ lorsque $p$ parcourt l’ensemble des nombres premiers.

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