On arithmetic progressions in finite fields (Q6045529)

A generator of the cyclic group \(\mathbb{F}^{*}_{q^{n}}\) of a finite field \(\mathbb{F}_{q^{n}}\) is called a primitive element. An element \(g\in \mathbb{F}_{q^{n}}\) is called normal if the elements \(g, g^{q}, \dots{} ,g^{q^{n-1}}\) form an \(\mathbb{F}_{q}\)-basis for \(\mathbb{F}_{q^{n}}\) as a vector space. The authors prove the existence of \(m\)-term arithmetic progressions with a given non-zero common difference in \(\mathbb{F}_{q^{n}}\) formed by primitive elements with at least one normal element in the following cases. For \(m = 4\) with char \((\mathbb{F}_{q}) \ge 5\), when \(q^{n} \ge 3.31\cdot 10^{2821}\) and for \(m \ge 5\) with char \((\mathbb{F}_{q}) \ge m\), when \(q > e^{17^{m+2}}\). The authors also prove the existence of such arithmetic progressions for \(m = 2\) and \(m = 3\) with some exceptional values of \(q\) and \(n\). They give the complete lists of exceptions as well. The authors prove some sieve-type estimations which ensure the existence of arithmetic progressions whose terms are primitive elements with at least one normal term among them. The proofs are based on the properties of some character sums. These results play a crucial role in the proofs.