Counting monic irreducible polynomials P in 𝔽 q [ X ] for which order of X ( mod P ) is odd

Christian Ballot[1]

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

Journal de Théorie des Nombres de Bordeaux (2007)

  • Volume: 19, Issue: 1, page 41-58
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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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 19.1 (2007): 41-58. <http://eudml.org/doc/249969>.

@article{Ballot2007,
abstract = {Hasse showed the existence and computed the Dirichlet density of the set of primes $p$ for which the order of $2\hspace\{4.44443pt\}(\@mod \; p)$ is odd; it is $7/24$. Here we mimic successfully Hasse’s method to compute the density $\delta _q$ of monic irreducibles $P$ in $\mathbb\{F\}_q[X]$ for which the order of $X\hspace\{4.44443pt\}(\@mod \; P)$ 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 $\delta _p$’s as $p$ varies through all rational primes.},
affiliation = {Département de Mathématiques, Université de Caen, Campus 2, 14032 Caen Cedex, France},
author = {Ballot, Christian},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {density; finite field},
language = {eng},
number = {1},
pages = {41-58},
publisher = {Université Bordeaux 1},
title = {Counting monic irreducible polynomials $P$ in $\{\mathbb\{F\}_q[X]\}$ for which order of $\{X\!\!\hspace\{4.44443pt\}(\@mod \; P)\}$ is odd},
url = {http://eudml.org/doc/249969},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Ballot, Christian
TI - Counting monic irreducible polynomials $P$ in ${\mathbb{F}_q[X]}$ for which order of ${X\!\!\hspace{4.44443pt}(\@mod \; P)}$ is odd
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 1
SP - 41
EP - 58
AB - Hasse showed the existence and computed the Dirichlet density of the set of primes $p$ for which the order of $2\hspace{4.44443pt}(\@mod \; p)$ is odd; it is $7/24$. Here we mimic successfully Hasse’s method to compute the density $\delta _q$ of monic irreducibles $P$ in $\mathbb{F}_q[X]$ for which the order of $X\hspace{4.44443pt}(\@mod \; P)$ 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 $\delta _p$’s as $p$ varies through all rational primes.
LA - eng
KW - density; finite field
UR - http://eudml.org/doc/249969
ER -

References

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