On the existence of primitive completely normal bases of finite fields

TL;DR

This paper proves the existence of primitive elements in finite field extensions that generate completely normal bases under certain conditions, advancing the understanding of finite field structure and basis construction.

Contribution

It establishes the existence of primitive completely normal bases in finite fields for specific degrees, extending previous theoretical results.

Findings

Existence of primitive completely normal bases proven for certain degrees

Conditions: n=p^{ ext{ell}}m, with (m,p)=1 and q>m

Provides theoretical foundation for basis construction in finite fields

Abstract

Let $\mathbb{F}_q$ be the finite field of characteristic $p$ with $q$ elements and $\mathbb{F}_{q^n}$ its extension of degree $n$. We prove that there exists a primitive element of $\mathbb{F}_{q^n}$ that produces a completely normal basis of $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$, provided that $n=p^{\ell}m$ with $(m,p)=1$ and $q>m$.