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 α in F q n such that α 3 − α + 1 is also a primitive element of F q n , a sufficient condition for the existence of a primitive normal element α in F q n over F q such that α 3 − α + 1 is a primitive element of F q n , and a sufficient condition for the existence of a primitive normal element α in F q n over F q such that α 3 − α + 1 is also a primitive normal element of F q n over F q. [ABSTRACT FROM AUTHOR]