A note on additive characters of finite fields

Abstract

Let ${\mathbb{F}}_q$ be the finite field with $q$ elements, where $q$ is a prime power and, for each integer $n\ge 1$, let ${\mathbb{F}}_{q^n}$ be the unique $n$-degree extension of ${\mathbb{F}}_q$. The ${\mathbb{F}}_q$-orders of an element in ${\mathbb{F}}_{q^n}$ and an additive character over ${\mathbb{F}}_{q^n}$ have been extensively used in the proof of existence results over finite fields (e.g., the Primitive Normal Basis Theorem). In this note we provide an interesting relation between these two objects.