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\(\mathbb{F}_q \).