A Permutation of a Small Infinite Field

Abstract

Let $\mathbb{F}$ denote the infinite field that is the union of all extensions of GF(2) of odd degree. Let q = 2^m and let ${\mathbb{F}_{Q}} = GF(2{q^{2}})$. We provide a simple proof of the permutation property of the polynomials x ^2q+1 + x ³ +x on ${\mathbb{F}_{Q}}$ for each m by viewing each member of the class as the action of a simply defined permutation of $\mathbb{F}$.