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arXiv · 1811.00896

Further results on the Morgan-Mullen conjecture

Theodoulos Garefalakis·Giorgos Kapetanakis
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Abstract

Let $\mathbb{F}_q$ be the finite field of characteristic $p$ with $q$ elements and $\mathbb{F}_{q^n}$ its extension of degree $n$. The conjecture of Morgan and Mullen asserts the existence of primitive and completely normal elements (PCN elements) for the extension $\mathbb{F}_{q^n}/\mathbb{F}_q$ for any $q$ and $n$. It is known that the conjecture holds for $n \leq q$. In this work we prove the conjecture for a larger range of exponents. In particular, we give sharper bounds for the number of completely normal elements and use them to prove asymptotic and effective existence results for $q\leq n\leq O(q^\epsilon)$, where $\epsilon=2$ for the asymptotic results and $\epsilon=1.25$ for the effective ones. For $n$ even we need to assume that $q-1\nmid n$.

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Theodoulos Garefalakis, Giorgos Kapetanakis. 2018-11-01. Further results on the Morgan-Mullen conjecture. https://arxiv.org/abs/1811.00896

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