Existence of primitive pairs with two prescribed traces over finite fields

Abstract

Given [Formula: see text], a field with [Formula: see text] elements, where [Formula: see text] is a prime power and [Formula: see text] is a positive integer. Let [Formula: see text] and [Formula: see text] is a rational function, where [Formula: see text] and [Formula: see text] are distinct irreducible polynomials with [Formula: see text] in [Formula: see text]. We construct a sufficient condition on [Formula: see text] which guarantees primitive pairing [Formula: see text] exists in [Formula: see text] such that [Formula: see text] and [Formula: see text] for any prescribed [Formula: see text]. Further, we demonstrate for any positive integer [Formula: see text], such a pair definitely exists for large [Formula: see text]. The scenario when [Formula: see text] is handled separately and we verified that such a pair exists for all [Formula: see text] except from possible 71 values of [Formula: see text]. A result for the case [Formula: see text] is given as well.