Primitive Normal Element and Its Inverse in Finite Fields

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