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: 2005-07-26
Published online: 2008-12-03
DOI: https://doi.org/10.5802/jtnb.572
@article{JTNB_2007__19_1_41_0,
     author = {Christian Ballot},
     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},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {1},
     year = {2007},
     pages = {41-58},
     doi = {10.5802/jtnb.572},
     zbl = {1142.11082},
     mrnumber = {2332052},
     language = {en},
     url={jtnb.centre-mersenne.org/item/JTNB_2007__19_1_41_0/}
}
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, Volume 19 (2007) no. 1, pp. 41-58. doi : 10.5802/jtnb.572. https://jtnb.centre-mersenne.org/item/JTNB_2007__19_1_41_0/

[Ba1] C. Ballot, Density of prime divisors of linear recurrences. Memoirs of the A.M.S., vol. 115, Nu. 551 (1995). | MR 1257079 | Zbl 0827.11006

[Ba2] C. Ballot, Competing prime asymptotic densities in 𝔽 q [X]. A discussion. Submitted preprint.

[Ba3] C. Ballot, An elementary method to compute prime densities in 𝔽 q [X]. To appear in Integers.

[Des] R. Descombes, Éléments de théorie des nombres. Presses Universitaires de France (1986). | MR 843073 | Zbl 0584.10001

[Ga] J. von zur Gathen et als, Average order in cyclic groups. J. Theor. Nombres Bordx, vol. 16, Nu. 1, (2004), 107–123. | Numdam | MR 2145575 | Zbl 1079.11003

[Ha] H. H. Hasse, Über die Dichte der Primzahlen p, für die eine vorgegebene ganzrationale Zahl a0 von gerader bzw. ungerader Ordnung mod p ist. Math. Annale 166 (1966), 19–23. | MR 205975 | Zbl 0139.27501

[Lag] J. C. Lagarias, The set of primes dividing the Lucas Numbers has density 2/3. Pacific J. Math., vol. 118, Nu. 2 (1985), 449–461 and “Errata”, vol. 162 (1994), 393–396. | Zbl 0790.11014

[Lan] S. Lang, Algebraic Number Theory. Springer-Verlag, 1986. | MR 1282723 | Zbl 0601.12001

[Lax] R. R. Laxton, Arithmetic Properties of Linear Recurrences. Computers and Number Theory (A.O.L. Atkin and B.J. Birch, Eds.), Academic Press, New York, 1971, 119–124. | Zbl 0226.10012

[M-S] P. Moree & P. Stevenhagen, Prime divisors of Lucas sequences. Acta Arithm., vol. 82, Nu. 4, (1997), 403–410. | MR 1483692 | Zbl 0913.11048

[Mo1] P. Moree, On the prime density of Lucas sequences. J. Theor. Nombres Bordx, vol. 8, Nu. 2, (1996), 449–459. | Numdam | MR 1438482 | Zbl 0873.11058

[Mo2] P. Moree, On the average number of elements in a finite field with order or index in a prescribed residue class. Finite fields Appl., vol. 10, Nu. 3, (2004), 438–463. | MR 2067608 | Zbl 1061.11050

[Nar] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers. PWN - Polish Scientific Publishers, 1974. | MR 347767 | Zbl 0276.12002

[Pra] K. Prachar, Primzahlverteilung. Springer-Verlag, 1957. | MR 87685 | Zbl 0080.25901

[Ro] M. Rosen, Number Theory in Function Fields. Springer-Verlag, Graduate texts in mathematics 210, 2002. | MR 1876657 | Zbl 1043.11079

[Ser] J. P. Serre, A course in Arithmetic. Springer-Verlag, 1973. | MR 344216 | Zbl 0432.10001

[Sier] W. Sierpinski, Sur une décomposition des nombres premiers en deux classes. Collect. Math., vol. 10, (1958), 81–83. | MR 103854 | Zbl 0084.27106

[Sti] H. Stichtenoth, Algebraic Function Fields and Codes. Springer-Verlag, 1993. | MR 1251961 | Zbl 0816.14011