Existence results for primitive elements in cubic and quartic extensions of a finite field
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- by Geoff Bailey, Stephen D. Cohen, Nicole Sutherland and Tim Trudgian HTML | PDF
- Math. Comp. 88 (2019), 931-947 Request permission
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 $n=4$.References
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Additional Information
- Geoff Bailey
- Affiliation: Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Camperdown NSW 2006, Australia
- MR Author ID: 618788
- Email: geoff.bailey@sydney.edu.au
- Stephen D. Cohen
- Affiliation: School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QQ, Scotland
- MR Author ID: 50360
- Email: stephen.cohen@glasgow.ac.uk
- Nicole Sutherland
- Affiliation: Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Camperdown NSW 2006, Australia
- MR Author ID: 975175
- 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
- MR Author ID: 909247
- Email: t.trudgian@adfa.edu.au
- 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.
- © Copyright 2018 American Mathematical Society
- Journal: Math. Comp. 88 (2019), 931-947
- MSC (2010): Primary 11T30, 11T06
- DOI: https://doi.org/10.1090/mcom/3357
- MathSciNet review: 3882289