Primitive Normal Polynomials Over Finite Fields
Ilene H. Morgan, Gary L. Mullen · Mathematics of Computation · 1994
In this note we significantly extend the range of published tables of primitive normal polynomials over finite fields. For each ${p^n} < {10^{50}}$ with $p \leq 97$, we provide a primitive normal polynomial of degree n over ${F_p}$. Moreover, each polynomial has the minimal number of nonzero coefficients among all primitive normal polynomials of degree n over ${F_p}$. The roots of such a polynomial generate a primitive normal basis of ${F_{{p^n}}}$ over ${F_p}$, and so are of importance in many computational problems. We also raise several conjectures concerning the distribution of such primitive normal polynomials, including a refinement of the primitive normal basis theorem.