Article contents
FINITE FIELD EXTENSIONS WITH THE LINE OR TRANSLATE PROPERTY FOR
$r$-PRIMITIVE ELEMENTS
Published online by Cambridge University Press: 02 March 2020
Abstract
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.
- Type
- Research Article
- Information
- Journal of the Australian Mathematical Society , Volume 111 , Issue 3 , 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.
- Copyright
- © 2020 Australian Mathematical Publishing Association Inc.
Footnotes
Communicated by I. Shparlinski
The first author is Emeritus Professor of Number Theory, University of Glasgow.
References
- 7
- Cited by