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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 .

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