A modification of Fitzgeraldʹs characterization of primitive polynomials over a finite field

A characterization of primitive polynomials, among irreducible polynomials, over a finite field is established. Such characterization is determined by the number of nonzero terms in certain quotient of polynomials. The proof, which is a modification of the one discovered by Fitzgerald in 2003 yet revealing more general structure, is based principally on counting the number of occurrences of elements in a linear recurring sequence over a finite field.