Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4583236 | Finite Fields and Their Applications | 2007 | 16 Pages |
In this paper, we prove that for any given nā©¾2, there exists a constant C(n) such that for any prime power q>C(n), there exists a primitive normal polynomial of degree n over Fq 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) 47ā61] 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) 506ā521; D. Mills, Existence of primitive polynomials with three coefficients prescribed, J. Algebra Number Theory Appl. 4 (2004) 1ā22] are the main tools to get the above result.