Variations of the Primitive Normal Basis Theorem

TL;DR

This paper explores variations of the Primitive Normal Basis Theorem, proving new results that confirm the existence of elements with specific properties in finite fields, thus advancing understanding in finite field theory.

Contribution

It completes the proof of a conjecture by Anderson and Mullen, establishing new existence results for elements with prescribed multiplicative order and trace.

Findings

Proved variations of the Primitive Normal Basis Theorem

Confirmed existence of elements with order (q^n-1)/2

Established elements with prescribed trace over finite fields

Abstract

The celebrated Primitive Normal Basis Theorem states that for any $n\ge 2$ and any finite field $\mathbb F_q$, there exists an element $\alpha\in \mathbb F_{q^n}$ that is simultaneously primitive and normal over $\mathbb F_q$. In this paper, we prove some variations of this result, completing the proof of a conjecture proposed by Anderson and Mullen (2014). Our results also imply the existence of elements of $\mathbb F_{q^n}$ with multiplicative order $(q^n-1)/2$ and prescribed trace over $\mathbb F_q$.