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Mathematics of Computation

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Existence results for primitive elements in cubic and quartic extensions of a finite field


Authors: Geoff Bailey, Stephen D. Cohen, Nicole Sutherland and Tim Trudgian
Journal: Math. Comp.
MSC (2010): Primary 11T30, 11T06
DOI: https://doi.org/10.1090/mcom/3357
Published electronically: May 18, 2018
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Abstract: With $ \mathbb{F}_q$ the finite field of $ q$ elements, we investigate the following question. If $ \gamma $ generates $ \mathbb{F}_{q^n}$ over $ \mathbb{F}_q$ and if $ \beta $ is a nonzero element of $ \mathbb{F}_{q^n}$, is there always an $ a \in \mathbb{F}_q$ such that $ \beta (\gamma + a)$ is a primitive element? We resolve this case when $ n=3$, thereby proving a conjecture by Cohen. We also substantially improve on what is known when $ n=4$.


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Additional Information

Geoff Bailey
Affiliation: Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Camperdown NSW 2006, Australia
Email: geoff.bailey@sydney.edu.au

Stephen D. Cohen
Affiliation: School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QQ, Scotland
Email: stephen.cohen@glasgow.ac.uk

Nicole Sutherland
Affiliation: Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Camperdown NSW 2006, Australia
Email: nicole.sutherland@sydney.edu.au

Tim Trudgian
Affiliation: School of Physical, Environmental and Mathematical Sciences, UNSW Canberra at the Australian Defence Force Academy, Campbell, ACT 2610, Australia
Email: t.trudgian@adfa.edu.au

DOI: https://doi.org/10.1090/mcom/3357
Keywords: Primitive elements, finite fields, cubic generators
Received by editor(s): July 8, 2017
Received by editor(s) in revised form: January 12, 2018
Published electronically: May 18, 2018
Additional Notes: The fourth author was supported by Australian Research Council Future Fellowship FT160100094.
Article copyright: © Copyright 2018 American Mathematical Society

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