Primitive Complete Normal Bases for Regular Extensions: Exceptional Cyclotomic Modules

Abstract

A primitive completely normal element for an extension $\mathbb{F}_{q^n}/\mathbb{F}_{q}$ of Galois fields is a generator of the multiplicative group of $\mathbb{F}_{q^n}$, which simultaneously is normal over every intermediate field of that extension. We are going to prove that such a generator exists when $\mathbb{F}_{q^n}/\mathbb{F}_{q}$ is an 'exceptional' regular extension. In combination with [6] our investigations altogether settle the existence of primitive completely normal bases for any regular extension. An important feature of the class of regular extensions is that they comprise every extension of prime power degree.