有限域上互反本原正规元的存在性

Primitive Normal Element and Its Inverse in Finite Fields

设q是素数方幂,n是正整数,Fqn是qn个元素的有限域．本文证明了:当正整数n≥32时,对任意的素数方幂q,存在Fqn中的本原元ξ满足ξ和ξ-1都是Fqn 在Fq上的正规元,也即{ξ,ξq,…,ξqn-1}和{ξ-1,ξ-q,…,ξ-qn-1)都构成Fqn在Fq 上的本原正规基．

Let q be a prime power, n be a positive integer, Fq^n denote a finite field with q^n elements. We prove that if n≥ 32, then there exists a primitive element ξof Fq^n satisfying that ξ and ξ^-1 are normal elements of Fq^n over Fq,ie,both{ζ-1,ξ-q,…,ξ-qn-1} and {ζ-1,ξ-q,…,ξ-qn-1}are primitive normal bases over Fq.