Given a prime power q and a positive integer n, let F q n denote the finite field with q n elements. Also let a, b be arbitrary members of the ground field F q . We investigate the existence of a non-zero element ξ ∈ F q n such that ξ + ξ −1 is primitive and T (ξ) = a, T (ξ −1 ) = b, where T (ξ) denotes the trace of ξ in F q . This was a question intended to be addressed by Cao and Wang in 2014. Their work dealt instead with another problem already in the literature. Our solution deals with all values of n ≥ 5.A related study involves the cubic extension F q 3 of F q . We show that when either q ≥ 8 • 10 12 or q 3 − 1 has at least 24 distinct prime factors, then, for any a ∈ F q we can find a primitive element ξ ∈ F q 3 such that ξ + ξ −1 is also a primitive element of F q 3 , and for which the trace of ξ is equal to a. The improves a result of Cohen and Gupta. Along the way we prove a hybridised lower bound on prime divisors in various residue classes, which may be of interest to related existence questions.
We prove that for any prime power
$q\notin \{3,4,5\}$
, the cubic extension
$\mathbb {F}_{q^{3}}$
of the finite field
$\mathbb {F}_{q}$
contains a primitive element
$\xi $
such that
$\xi +\xi ^{-1}$
is also primitive, and
$\operatorname {\mathrm {Tr}}_{\mathbb {F}_{q^{3}}/\mathbb {F}_{q}}(\xi )=a$
for any prescribed
$a\in \mathbb {F}_{q}$
. This completes the proof of a conjecture of Gupta et al. [‘Primitive element pairs with one prescribed trace over a finite field’, Finite Fields Appl.54 (2018), 1–14] concerning the analogous problem over an extension of arbitrary degree
$n\ge 3$
.
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