Variations of the Primitive Normal Basis Theorem

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 ${\mathbb{F}}_q$.

19 pages