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An extension of the (strong) primitive normal basis theorem

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Abstract

An extension of the primitive normal basis theorem and its strong version is proved. Namely, we show that for nearly all \(A = {\small \left( \begin{array}{cc} a&{}b \\ c&{}d \end{array} \right) } \in \mathrm{GL}_2(\mathbb {F}_{q})\), there exists some \(x\in \mathbb {F}_{q^m}\) such that both \(x\) and \((-dx+b)/(cx-a)\) are simultaneously primitive elements of \(\mathbb {F}_{q^m}\) and produce a normal basis of \(\mathbb {F}_{q^m}\) over \(\mathbb {F}_q\), granted that \(q\) and \(m\) are large enough.

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Acknowledgments

I would like to thank my supervisor, Prof. Theodoulos Garefalakis, for pointing out this problem to me and for his useful suggestions. I would also like to thank my friend Maria Chlouveraki for her comments and Prof. Stephen D. Cohen for pointing out a serious mistake in the manuscripts and for his comments. Finally, I wish to thank the anonymous reviewers for their suggestions, that vastly improved the quality of this paper. This work was supported by the University of Crete’s research Grant No. 3744.

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  1. Department of Mathematics and Applied Mathematics, University of Crete, Voutes Campus, 70013 , Heraklion, Crete, Greece

    Giorgos Kapetanakis

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  1. Giorgos Kapetanakis
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Correspondence to Giorgos Kapetanakis.

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Kapetanakis, G. An extension of the (strong) primitive normal basis theorem. AAECC 25, 311–337 (2014). https://doi.org/10.1007/s00200-014-0230-7

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  • DOI: https://doi.org/10.1007/s00200-014-0230-7

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