Arithmetic progressions of some special elements in finite fields

Abstract

Let q be a prime power and n be a positive integer. Let Fqn be the degree n field extension over the field Fq. In this paper, we study the existence of r-primitive elements in an arithmetic progression of length m >= 2. We find a condition for the existence of an element alpha is an element of Fqnx for a given m >= 2 and gamma is an element of Fqnx such that, for 1 <= i <= m, alpha+(i-1)gamma is an element of Fqnx is ri-primitive with prescribed norm ai is an element of Fqx. Further, we improve this sufficient condition when q equivalent to 3 (mod4) and the positive integer n is odd. Also, for n >= 7,m=2 we demonstrate that there are only 41 possible exceptions, and at most 2 exceptions when q equivalent to 3 (mod4) and n is odd.