Primitive normal polynomials with the first two coefficients prescribed: A revised p-adic method

Abstract

Let F q be a finite field with q = p k elements. We prove that for any given n ⩾ 7 , and any elements a , b ∈ F q , a ≠ 0 , there exists a primitive normal polynomial f ( x ) of degree n, f ( x ) = x n − σ 1 x n − 1 + ⋯ + ( − 1 ) n σ n , with the first two coefficients σ 1 , σ 2 prescribed as a, b, respectively. This result strengthens the results of the existence of primitive polynomials with two coefficients prescribed [S.D. Cohen, D. Mills, Primitive polynomials with the first and second coefficients prescribed, Finite Fields Appl. 9 (2003) 334–350; W.B. Han, The coefficients of primitive polynomials over finite fields, Math. Comp. 65 (1996) 331–340; W.B. Han, On two exponential sums and their applications, Finite Fields Appl. 3 (1997) 115–130] and the existence of primitive normal bases with prescribed trace [S.D. Cohen, D. Hachenberger, Primitive normal bases with prescribed trace, Appl. Algebra Engrg. Comm. Comput. 9 (1999) 383–403]. In order to use the p-adic method which proposed by the first two authors, we first lift the definition of both primitive and normal from finite fields to Galois rings. Then we discuss the existence of lifted primitive normal polynomials over Galois rings and finally establish the existence of primitive normal polynomials with the first two coefficients prescribed with the help of character sums over Galois rings and Cohen's various sieve techniques. In order to make this paper more succinct, we deal with the case n = 4 , 5 , 6 in another paper since the computation is more miscellaneous and complicated.