Dickson Polynomials, Completely Normal Polynomials and the Cyclic Module Structure of Specific Extensions of Finite Fields

Abstract

Let q be a prime power and n be a product of odd prime factors of (q+1). We exhibit a description of the irreducible, cyclic ${\mathbb{F}}_q [X]$-submodules of ${\mathbb{F}}_{q^n}$ in terms of the roots of an irreducible polynomial D _n (X,a)-b where D _n (X,a) is the n-th Dickson polynomial of the first kind. With the help of this description it is examined for which $s,t \in {\mathbb{F}}_q$ the inverses of the roots of D _n (sX-t,a)-b constitute normal bases over every intermediate subfield ${\mathbb{F}}_{q^d}$ between ${\mathbb{F}}_{q^n}$ and ${\mathbb{F}}_q$. This leads to some new specific constructions of completely normal polynomials.