A conjecture on primitive polynomials over finite fields

We present a conjecture about a family of polynomials over a finite field that appears to always contain a primitive polynomial. Cohen has shown that given α ∈ F q 2 such that α ∉ F q , there exists λ ∈ F q such that λ − α is a primitive element in F q 2 ⁎ . Following considerations of certain rank 2 Drinfeld modules, we present a conjecture that given any nonzero μ ∈ F q , there exists λ ∈ F q such that x 2 + μ x + λ − α is a primitive quadratic polynomial in F q 2 x , for q > 43 . The conjecture has an equivalent statement in terms of quartic polynomials over F q .