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

Andrew R Booker; Stephen Cohen; Nicol Leong; Tim Trudgian

doi:10.1017/S0004972722000442

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

Andrew R Booker, Stephen Cohen, Nicol Leong, Tim Trudgian

School of Mathematics

Research output: Contribution to journal › Article (Academic Journal) › peer-review

3 Citations (Scopus)

63 Downloads (Pure)

Abstract

We prove that for any prime power q∉{3,4,5} , the cubic extension Fq3 of the finite field Fq contains a primitive element ξ such that ξ+ξ−1 is also primitive, and TrFq3/Fq(ξ)=a for any prescribed a∈Fq . This completes the proof of a conjecture of Gupta et al. [‘Primitive element pairs with one prescribed trace over a finite field’, Finite Fields Appl.54 (2018), 1–14] concerning the analogous problem over an extension of arbitrary degree n≥3 .

Original language	English
Pages (from-to)	458-462
Number of pages	5
Journal	Bulletin of the Australian Mathematical Society
Volume	106
Issue number	3
Early online date	25 Apr 2022
DOIs	https://doi.org/10.1017/S0004972722000442
Publication status	E-pub ahead of print - 25 Apr 2022

Bibliographical note

Funding Information:
T. Trudgian was supported by Australian Research Council Future Fellowship FT160100094.

Publisher Copyright:
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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10.1017/S0004972722000442

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This is the accepted author manuscript (AAM). The final published version (version of record) is available online via Cambridge University Press at https://doi.org/10.1017/S0004972722000442. Please refer to any applicable terms of use of the publisher.
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title = "Primitive element pairs with a prescribed trace in the cubic extension of a finite field",

abstract = "We prove that for any prime power q∉\{3,4,5\} , the cubic extension Fq3 of the finite field Fq contains a primitive element ξ such that ξ+ξ−1 is also primitive, and TrFq3/Fq(ξ)=a for any prescribed a∈Fq . This completes the proof of a conjecture of Gupta et al. [{\textquoteleft}Primitive element pairs with one prescribed trace over a finite field{\textquoteright}, Finite Fields Appl.54 (2018), 1–14] concerning the analogous problem over an extension of arbitrary degree n≥3 .",

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AU - Booker, Andrew R

AU - Cohen, Stephen

AU - Leong, Nicol

AU - Trudgian, Tim

N1 - Funding Information: T. Trudgian was supported by Australian Research Council Future Fellowship FT160100094. Publisher Copyright: © The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

PY - 2022/4/25

Y1 - 2022/4/25

N2 - We prove that for any prime power q∉{3,4,5} , the cubic extension Fq3 of the finite field Fq contains a primitive element ξ such that ξ+ξ−1 is also primitive, and TrFq3/Fq(ξ)=a for any prescribed a∈Fq . This completes the proof of a conjecture of Gupta et al. [‘Primitive element pairs with one prescribed trace over a finite field’, Finite Fields Appl.54 (2018), 1–14] concerning the analogous problem over an extension of arbitrary degree n≥3 .

AB - We prove that for any prime power q∉{3,4,5} , the cubic extension Fq3 of the finite field Fq contains a primitive element ξ such that ξ+ξ−1 is also primitive, and TrFq3/Fq(ξ)=a for any prescribed a∈Fq . This completes the proof of a conjecture of Gupta et al. [‘Primitive element pairs with one prescribed trace over a finite field’, Finite Fields Appl.54 (2018), 1–14] concerning the analogous problem over an extension of arbitrary degree n≥3 .

U2 - 10.1017/S0004972722000442

DO - 10.1017/S0004972722000442

M3 - Article (Academic Journal)

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SP - 458

EP - 462

JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

IS - 3

ER -

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

Abstract

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