On Existence of Primitive Normal Elements of Cubic Form over Finite Fields

For a prime p and a positive integer k, let q = p(k) and F-q(n) be the extension field of F-q. We derive a sufficient condition for the existence of a primitive element a in F-q(n) such that alpha(3) - alpha + 1 is also a primitive element of F-q(n), a sufficient condition for the existence of a primitive normal element alpha in F-q(n) over F-q such that alpha(3) - alpha+1 is a primitive element of F-q(n), and a sufficient condition for the existence of a primitive normal element a in F-q(n) over F-q such that alpha(3) - alpha + 1 is also a primitive normal element of F-q(n) over F-q.