Equidistribution of Primitive Normal Elements in Finite Fields

Abstract

Let $q=p^k$ be a prime power, let $n\ge 2$ be an integer and let ${\mathbb{F}}_{q^n}$ be a finite field. It is shown that the set of primitive normal elements is a Salem set. Furthermore, it is proved that this set is strongly equidistributed in the finite field. Similar results are proved for the set of quadratic residues and the set of primitive roots modulo a large prime $p\ge 3$.

Seventeen Pages. Keywords: Finite field; Quadratic Residues; Primitive element; Normal element; primitive normal element; Salem set; Equidistribution; Finite Fourier transform