Further results on the Morgan-Mullen conjecture

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 $q-1\nmid n$.Comment: arXiv admin note: text overlap with arXiv:1709.0314

Transitive Lie algebras of vector fields---an overview

This overview paper is intended as a quick introduction to Lie algebras of vector fields. Originally introduced in the late 19th century by Sophus Lie to capture symmetries of ordinary differential equations, these algebras, or infinitesimal groups, are a recurring theme in 20th-century research on Lie algebras. I will focus on so-called transitive or even primitive Lie algebras, and explain their theory due to Lie, Morozov, Dynkin, Guillemin, Sternberg, Blattner, and others. This paper gives just one, subjective overview of the subject, without trying to be exhaustive.Comment: 20 pages, written after the Oberwolfach mini-workshop "Algebraic and Analytic Techniques for Polynomial Vector Fields", December 2010 2nd version, some minor typo's corrected and some references adde

On the existence of primitive completely normal bases of finite fields

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 $q>m$