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A class of permutation trinomials over finite fields

Xiang-dong Hou (2014)

Acta Arithmetica

Let q > 2 be a prime power and f = - x + t x q + x 2 q - 1 , where t * q . We prove that f is a permutation polynomial of q ² if and only if one of the following occurs: (i) q is even and T r q / 2 ( 1 / t ) = 0 ; (ii) q ≡ 1 (mod 8) and t² = -2.

Counting irreducible polynomials over finite fields

Qichun Wang, Haibin Kan (2010)

Czechoslovak Mathematical Journal

In this paper we generalize the method used to prove the Prime Number Theorem to deal with finite fields, and prove the following theorem: π ( x ) = q q - 1 x log q x + q ( q - 1 ) 2 x log q 2 x + O x log q 3 x , x = q n where π ( x ) denotes the number of monic irreducible polynomials in F q [ t ] with norm x .

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.

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