Primitive Complete Normal Bases for Regular Extensions: Exceptional Cyclotomic Modules

TL;DR

This paper proves the existence of primitive completely normal elements in certain regular extensions of finite fields, specifically 'exceptional' cases, completing the understanding of such bases for all regular extensions.

Contribution

It establishes the existence of primitive completely normal bases for all regular extensions, including exceptional cases, advancing finite field theory.

Findings

Existence of primitive completely normal elements in exceptional regular extensions.

Complete characterization of primitive completely normal bases for all regular extensions.

Resolution of the problem for extensions of prime power degree.

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.