Primitive element pairs with a prescribed trace in the cubic extension of a finite field

We prove that for any prime power \(q\notin\{3,4,5\}\), the cubic extension \(\mathbb{F}_{q^3}\) of the finite field \(\mathbb{F}_q\) contains a primitive element \(\xi\) such that \(\xi+\xi^{-1}\) is also primitive, and \(\textrm{Tr}_{\mathbb{F}_{q^3}/\mathbb{F}_q}(\xi)=a\) for any prescribed \(a\i...

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Bibliographic Details
Published in:arXiv.org 2022-02
Main Authors: Booker, Andrew R, Cohen, Stephen D, Leong, Nicol, Trudgian, Tim
Format: Article
Language:English
Subjects:
Fields (mathematics)
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Summary:We prove that for any prime power \(q\notin\{3,4,5\}\), the cubic extension \(\mathbb{F}_{q^3}\) of the finite field \(\mathbb{F}_q\) contains a primitive element \(\xi\) such that \(\xi+\xi^{-1}\) is also primitive, and \(\textrm{Tr}_{\mathbb{F}_{q^3}/\mathbb{F}_q}(\xi)=a\) for any prescribed \(a\in\mathbb{F}_q\). This completes the proof of a conjecture of Gupta, Sharma, and Cohen concerning the analogous problem over an extension of arbitrary degree \(n\ge3\).
ISSN:2331-8422