本原正规多项式前两项系数的研究

Research on the First Two Coefficients of Primitive Normal Polynomials

设Fq表示q个元素的有限域,q为素数方幂。对n≥7,文献证明了存在Fq上n次本原正规多项式f(x)=xn-σ1xn-1+σ2xn-2+…+(-1)nσn,使得其前两项系数σ1(≠0),σ2可预先任意给定。文章讨论了剩余的n=5,6两种情况,通过使用Cohen筛法的新形式,改进了文献中的计算,从而将结论推广到n≥5。

Let Fq be a finite field with q elements,where q is a prime power.It has been proved that there exists a primitive normal polynomial f（x）=x^nσ1x^n-1＋σ2x^n-2＋…＋（-1）^nσn with the first two ecofficients σ1（≠0）,σ2 prescribed for n≥7 in documents.In this paper,the result of documents is extended to n≥5 with some new variation of the Cohen＇s sieve techniques.