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ON THE PRIMITIVITY OF THE TRINOMIAL $x^{n}+ax^{k}+b$ OVER $\mathbb {F}_{q}$

Part of: Algorithms - Computer Science Finite fields and commutative rings (number-theoretic aspects) Computational aspects of field theory and polynomials

Published online by Cambridge University Press:  11 September 2025

SOUFYANE BOUGUEBRINE Open the ORCID record for SOUFYANE BOUGUEBRINE [Opens in a new window]  and
RACHID BOUMAHDI Open the ORCID record for RACHID BOUMAHDI [Opens in a new window]
SOUFYANE BOUGUEBRINE
Affiliation:
https://ror.org/02xv2bn88 National Higher School of Mathematics , Sidi Abdellah, Algiers, Algeria e-mail: soufyane.bouguebrine@nhsm.edu.dz
RACHID BOUMAHDI*
Affiliation:
https://ror.org/02xv2bn88 National Higher School of Mathematics , Sidi Abdellah, Algiers, Algeria
*
e-mail: rachid.boumahdi@nhsm.edu.dz
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Abstract

This note provides an alternative proof of a theorem by Li et al. [‘On the primitivity of some trinomials over finite fields’, Adv. Math. (China) 44(3) (2015), 387–393] regarding the nonprimitivity of the trinomial $x^{n}+ax+b$ over $\mathbb {F}_{q^{m}}$ under the condition $a^{n}b^{1-n}\in \mathbb {F}_{q^{u}}^{\ast }$ for some positive integer $u<m$. We extend this result to the trinomial $x^{n}+a^{k}x^{k}+b^{k}$, showing its nonprimitivity over $\mathbb {F}_{q^{m}}$ when $ a^{n}b^{k-n}\in \mathbb {F}_{q^{u}}^{\ast }$ for some positive integer $u<m$. While the existing proof relies on the theory of linear recurrences over finite fields, our approach is short and self-contained, requiring no prior knowledge of this area.

Keywords

finite fieldprimitive polynomialordertrinomial

MSC classification

Primary: 11T06: Polynomials
Secondary: 12Y05: Computational aspects of field theory and polynomials 68W30: Symbolic computation and algebraic computation

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 5
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc

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References

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