On Primitive Normal Elements Over Finite Fields

Let q be a prime power, n a positive integer and F_qⁿ a finite field with qⁿ elements. In the thesis, given a, b∈F_q^* we study whether there exists an element ζ in F_qⁿ which satisfies following two or three conditions:(1) ζ is a primitive element in F_qⁿ;(2) both ζ and ζ^-1 are normal elements of F_qⁿ over F_q, i.e., both {ζ, ζ^q,...,ζ^qn-1}and{ζ^-1,ζ^-q,...,ζ^-q(n-1)} are basis over F_q.(3) T_rFqⁿ/fq(ζ) =a and T_rFqⁿ/fq(ζ^-1)=bAn element ζ in F_qⁿ is called a primitive nomal element over F_q if it is a primitive element in F_qⁿ as well as a nomal element of F_qⁿ over F_q. In Section 2 and Section 3 of Chapter 2, we respectively prove that1. If n ≥ 32, then for any prime power q, there exists a primitive normal element ζ of F_qⁿ over F_q such that ζ^-1 is also a primitive normal element of F_qⁿ over F_q.2. If n ≥ 32 and (q, n)(?){(2, 45), (2, 63), (3, 32), (3, 40), (7, 48), (9, 40), (19, 36), (37,36), (41,40)}, then for any a, b ∈ F_q^*, there exists a primitive normal element ζ of F_qⁿ over F_q such that ζ^-1 is also a primitive normal element of F_qⁿ over F_q, T_rFqⁿ/fq(ζ)= a and T_rFqⁿ/fq(ζ^-1)= b.