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# FINITE FIELD EXTENSIONS WITH THE LINE OR TRANSLATE PROPERTY FOR $r$-PRIMITIVE ELEMENTS

Published online by Cambridge University Press:  02 March 2020

## Abstract

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Let $r,n>1$ be integers and $q$ be any prime power $q$ such that $r\mid q^{n}-1$ . We say that the extension $\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$ possesses the line property for $r$ -primitive elements property if, for every $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}\in \mathbb{F}_{q^{n}}^{\ast }$ such that $\mathbb{F}_{q^{n}}=\mathbb{F}_{q}(\unicode[STIX]{x1D703})$ , there exists some $x\in \mathbb{F}_{q}$ such that $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D703}+x)$ has multiplicative order $(q^{n}-1)/r$ . We prove that, for sufficiently large prime powers $q$ , $\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$ possesses the line property for $r$ -primitive elements. We also discuss the (weaker) translate property for extensions.

## MSC classification

Type
Research Article
Information
Journal of the Australian Mathematical Society , December 2021 , pp. 313 - 319
Creative Commons
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
© 2020 Australian Mathematical Publishing Association Inc.

## Footnotes

Communicated by I. Shparlinski

The first author is Emeritus Professor of Number Theory, University of Glasgow.

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