Arithmetic progression in a finite field with prescribed norms

Abstract

Given a prime power q and a positive integer n, let 𝔽 q n {\mathbb{F}_{q^{n}}} represent a finite extension of degree n of the finite field 𝔽 q {{\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 ≥ 6 {n\geq 6} , q = 3 k {q=3^{k}} , m = 2 {m=2} we establish that there are only 10 possible exceptions.