On the existence of some specific elements in finite fields of characteristic 2

Abstract

Let q be a power of 2, n be a positive integer, and let ${\mathbb{F}}_{q^n}$ be the finite field with $q^n$ elements. In this paper, we consider the existence of some specific elements in ${\mathbb{F}}_{q^n}$. The main results obtained in this paper are listed as follows:

• There is an element ξ in ${\mathbb{F}}_{q^n}$ such that both ξ and $\xi +{\xi }^{-1}$ are primitive elements of ${\mathbb{F}}_{q^n}$ if $q=2^s$, and n is an odd number no less than 13 and $s>4$.

• For $q=2^s$, and any odd n, there is an element ξ in ${\mathbb{F}}_{q^n}$ such that ξ is a primitive normal element and $\xi +{\xi }^{-1}$ is a primitive element of ${\mathbb{F}}_{q^n}$ if either $n|(q-1)$, and $n⩾33$, or $n\nmid (q-1)$, and $n⩾30$, $s⩾6$.