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

Part of: Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  02 March 2020

STEPHEN D. COHEN Open the ORCID record for STEPHEN D. COHEN [Opens in a new window]  and
GIORGOS KAPETANAKIS Open the ORCID record for GIORGOS KAPETANAKIS [Opens in a new window]
STEPHEN D. COHEN
Affiliation:
6 Bracken Road, Portlethen, AberdeenAB12 4TA, UK e-mail: [email protected]
GIORGOS KAPETANAKIS*
Affiliation:
Department of Mathematics and Applied Mathematics, University of Crete, Voutes Campus, 70013Heraklion, Greece
*
e-mail: [email protected]
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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.

Keywords

primitive elementhigh-order elementline propertytranslate property

MSC classification

Primary: 11T30: Structure theory
Secondary: 11T06: Polynomials
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 111 , Issue 3 , December 2021 , pp. 313 - 319
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Creative Commons
Creative Common License - CCCreative Common License - BY
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.
Copyright
© 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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