An extension of the (strong) primitive normal basis theorem

Original Paper
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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.

Keywords

Finite field Primitive element Normal basis 

Mathematics Subject Classification (2010)

11T30 11T06 11T24 12E20 
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Notes

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsUniversity of CreteHeraklion, CreteGreece

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