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

AbstractLet 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 Dn(X,a)-b where Dn(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 Dn(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.