On arithmetic progressions in finite fields

In this paper, we explore the existence of m-term arithmetic progressions in $${\mathbb {F}}_{q^n}$$ with a given common difference whose terms are all primitive elements, and at least one them is normal. We obtain asymptotic results for $$m \ge 4$$ concrete \in \{2,3\}$$ , where complete list exceptions when belongs to {F}}_{q}$$ obtained. The proofs combine character sums, sieve estimates computational arguments using software SageMath.