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. **88** (2019), 931-947

MSC (2010):
Primary 11T30, 11T06

DOI:
https://doi.org/10.1090/mcom/3357

Published electronically:
May 18, 2018

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Abstract | References | Similar Articles | Additional Information

Abstract: With the finite field of elements, we investigate the following question. If generates over and if is a nonzero element of , is there always an such that is a primitive element? We resolve this case when , thereby proving a conjecture by Cohen. We also substantially improve on what is known when .

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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