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

TL;DR

This paper investigates the multiplicative orders of elements of the form 1 + 1^{-1} in finite fields of characteristic two, building on graph structures related to the map x 7 x^{-1}.

Contribution

It provides new results on the possible multiplicative orders of 1 + 1^{-1} in characteristic two finite fields, extending previous graph-based analyses.

Findings

Identifies constraints on the multiplicative orders of 1 + 1^{-1} in characteristic two

Relates the order properties to the structure of associated graphs

Advances understanding of element behavior in finite fields of characteristic two

Abstract

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.