Abstract
Let \(\mathbb {F}_{q^n}\) be a finite field with \(q^n\) elements, and let \(m_1\) and \(m_2\) be positive integers. Given polynomials \(f_1(x), f_2(x) \in \mathbb {F}_{q^n}[x]\) with \(\deg (f_i(x)) \le m_i\), for \(i = 1, 2\), and such that the rational function \(f_1(x)/f_2(x)\) satisfies certain conditions which we define, we present a sufficient condition for the existence of a primitive element \(\alpha \in \mathbb {F}_{q^n}\), normal over \(\mathbb {F}_q\), such that \(f_1(\alpha )/f_2(\alpha )\) is also primitive.
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Acknowledgements
Cícero Carvalho was partially funded by FAPEMIG APQ-01645-16, João Paulo Guardieiro was partially funded by CAPES 88882.441370/2019-01, Victor G.L. Neumann was partially funded by FAPEMIG APQ-03518-18 and Guilherme Tizziotti was partially funded by CNPq 307037/2019-3.
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Carvalho, C., Guardieiro, J.P., Neumann, V.G.L. et al. On the Existence of Pairs of Primitive and Normal Elements Over Finite Fields. Bull Braz Math Soc, New Series 53, 677–699 (2022). https://doi.org/10.1007/s00574-021-00277-2
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DOI: https://doi.org/10.1007/s00574-021-00277-2