有限域上本原多项式系数的研究

THE COEFFICIENTS OF PRIMITIVE POLYNOMIAL OVER FINITE FIELD

设Fq表示有q个元素的有限域,q为素数的方幂,f(x)=xn+a1xn-1+…+an-1x+an∈Fq[x].当n≥7时,文[8]指出存在Fq上可预先指定a1,a2的n次本原多项式.本文讨论了剩余的n=5,6两种情形,利用有限域上的两类特征和估计及Cohen筛法(见[4,6]),改进了文[8]中关于本原解个数的下界,并得到当n=5,6时,在特征为奇的有限域上存在可预先指定前两项系数的n次本原多项式.

Let f(x) = xn + a1xn-1+…+an-1x + an be a polynomial over Fq, where Fq denotes the finite field of q elements, q an odd prime power. When n ≥ 7, Han has stated that there exist a primitive polynomial of degree n with a1, a2 prescribed. In this paper, the authors discuss the remaining cases when n = 5,6. By making use of two kinds of exponential sums and Cohen's Sieve, the authors improve the lower bound of primitive solutions and prove that there exists a primitive polynomial of degree n = 5,6 with a1,a2 prescribed.