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Abstract: In this paper, we established the existence of a primitive normal polynomial over any finite field with any specified coefficient arbitrarily prescribed. Let be a positive integer and q a prime power. We prove that for any and any , there exists a primitive normal polynomial such that , with the only exceptions . The theory can be extended to polynomials of smaller degree too. [Copyright &y& Elsevier]

Abstract: In this paper, we prove that for any given , there exists a constant such that for any prime power , there exists a primitive normal polynomial of degree n over with the last coefficients prescribed, where the last coefficient is a primitive element. Furthermore, the number of prescribed coefficients increases from to when the coefficients are specified as with b any primitive element. This result is a complement to the existence of a primitive normal polynomial with the first coefficients prescribed which was proved in [S.Q. Fan, W.B. Han, K.Q. Feng, Primitive normal polynomials with multiple coefficients prescribed: An asymptotic result, Finite Fields Appl. 13 (2007) 1029–1044]. The outline of this paper is similar to the above reference with the following two different treatments. On one hand, we use instead of x in the problem reduction step and as a consequence use the hybrid kloostermann sums instead of hybrid weil sums over Galois rings. On the other hand, the estimates are slightly more complicated and the results in some special cases are better than those in [S.Q. Fan, W.B. Han, K.Q. Feng, Primitive normal polynomials with multiple coefficients prescribed: An asymptotic result, Finite Fields Appl. 13 (2007) 1029–1044]. [Copyright &y& Elsevier]

Abstract: In this paper, we prove that for any given , there exists a constant such that for any prime power , there exists a primitive normal polynomial of degree n over with the first coefficients prescribed, where the first coefficient is nonzero. This result strengthens the asymptotic result of the existence of primitive polynomials with the first coefficients prescribed [S.Q. Fan, W.B. Han, p-Adic formal series and Cohen''s problem, Glasg. Math. J. 46 (2004) 47–61] in two aspects. One is that we discuss in this paper not only the primitivity but also the normality. Another is that the number of the prescribed coefficients increases from to . The estimates of character sums over Galois rings, the p-adic method introduced by the first two authors, and the computation technique used in [S.Q. Fan, W.B. Han, Primitive polynomial with three coefficients prescribed, Finite Fields Appl. 10 (2004) 506–521; D. Mills, Existence of primitive polynomials with three coefficients prescribed, J. Algebra Number Theory Appl. 4 (2004) 1–22] are the main tools to get the above result. [Copyright &y& Elsevier]

Abstract: Let be a finite field with elements. We prove that for any given , and any elements , , there exists a primitive normal polynomial of degree n, , with the first two coefficients , 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 in another paper since the computation is more miscellaneous and complicated. [Copyright &y& Elsevier]

The authors proved in Fan and Han (Finite Field Appl., in press) that, for any given (a1,a2,a3)∈Fq3, there exists a primitive polynomial f(x)=xn-σ1xn-1+⋯+(-1)nσn over Fq of degree n with the first three coefficients σ1,σ2,σ3 prescribed as a1,a2,a3 when n⩾8. But the methods in Fan and Han (in press) are not effective for the case of n=7. Mills (Existence of primitive polynomials with three coefficients prescribed, J. Algebra Number Theory Appl., in press) resolves the n=7 case for finite fields of characteristic at least 5. In this paper, we deal with the remaining cases and prove that there exists a primitive polynomial of degree 7 over Fq with the first three coefficient prescribed where the characteristic of Fq is 2 or 3. [Copyright &y& Elsevier]