Arithmetic progression in a finite field with prescribed norms

Abstract

Given a prime power $q$ and a positive integer $n$, let ${\mathbb{F}}_{q^n}$ represents a finite extension of degree $n$ of the finite field ${\mathbb{F}}_q$. In this article, we investigate the existence of $m$ elements in arithmetic progression, where every element is primitive and at least one is normal with prescribed norms. Moreover, for $n\ge 6,q=3^k,m=2$ we establish that there are only $10$ possible exceptions.