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Counting monic irreducible polynomials $P$ in $𝔽_{q} [X]$ for which order of $X (mod P)$ is odd

Christian Ballot (2007)

Journal de Théorie des Nombres de Bordeaux

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.

Displaying 21 – 32 of 32

Counting monic irreducible polynomials $P$ in $𝔽_{q} [X]$ for which order of $X (mod P)$ is odd

Counting optimal joint digit expansions.

Counting points in medium characteristic using Kedlaya's algorithm.

Coverings of Singular curves over Finite Fields.

Critères d'irréductibilite des polynômes composés à coefficients dans un corps fini

Cube-Tilings of Rn and Nonlinear Codes.

Cycles of polynomials in algebraically closed fields of positive characteristic

Cyclic Boolean algebras and Galois fields $G F (2^{k})$

Cyclotomic numbers of order 2l, l an odd prime

Cyclotomic polynomials with large coefficients

Cyclotomic properties of the Rudin-Shapiro polynomials.

Cyclotomy of order 15 over $G F (p^{2})$ , $p = 4, 11$ (mod 15).

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