Some notes on the multiplicative order of $\alpha + \alpha^{-1}$ in finite fields of characteristic two

In this paper we prove some results on the possible multiplicative orders of $\alpha + \alpha^{-1}$ when $\alpha$ is a non-zero element of a finite field of characteristic 2. The results of the paper rely on a previous investigation on the structure of the graphs associated with the map $x \mapsto x + x^{-1}$ in finite fields of characteristic 2.