On primitive and free roots in a finite field

Abstract

In this paper them-dimensional extension$\mathbb{F}_{q^m }$ of the finite field$\mathbb{F}_q$ of orderq is investigated from an algebraic point of view. Looking upon the additive group$(\mathbb{F}_{q^m } , + )$ as a cyclic module over the principal ideal domain$\mathbb{F}_q [x]$, we introduce a new family of polynomials over$\mathbb{F}_q$ which are the additive analogues of the cyclotomic polynomials. Two methods to calculate these polynomials are proposed. In combination with algorithms to compute cyclotomic polynomials, we obtain, at least theoretically, a method to determine all elements in$\mathbb{F}_{q^m }$ of a given additive and multiplicative order; especially the generators of both cyclic structures, namely the generators of primitive normal bases in$\mathbb{F}_{q^m }$ over$\mathbb{F}_q$, are characterized as the set of roots of a certain polynomial over$\mathbb{F}_q$.