On the Existence of Some Specific Primitive Elements over Finite Fields of Even Characteristic

Let q be a power of 2, n be a positive integer, F_qn denote a finite field with qⁿ elements. In this paper, we consider the existence of the some specific elements in F_qn. The main results obtained in this paper are listed as follows: There is an element in F_qn such that both ξ and ξ+ξ^-1 are primitive elements of F_qn and ξ + ξ^-1 is also a normal element of F_qn if 1) either n|(q - 1), and n ≥ 37, or 2) n|(q - 1), and n ≥ 34, s ≥ 6.