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.
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Scheerhorn, A. Dickson Polynomials, Completely Normal Polynomials and the Cyclic Module Structure of Specific Extensions of Finite Fields. Designs, Codes and Cryptography 9, 193–202 (1996). https://doi.org/10.1023/A:1018070131208
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DOI: https://doi.org/10.1023/A:1018070131208