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Ovoids and primitive normal bases for quartic extensions of Galois fields

  • Dirk HachenbergerEmail author
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Abstract

We determine lower bounds for the number of primitive normal elements in a four-dimensional extension E over a Galois field \(F={\mathrm{GF}}(q)\). Our approach is based on viewing E as the three-dimensional projective space \(\Gamma =\mathrm{PG}(3,q)\). In any of the three cases, whether q is even, or \(q\equiv 3 { \text{ mod } }4\), or \(q\equiv 1 { \text{ mod } }4\), we use a decomposition of the multiplicative group of E in order to determine a (canonical) partition of the point set of \(\Gamma \) into \(q+1\) ovoids. The points of \(\Gamma \) are distinguished into primitive and non-primitive ones, and an ovoid is called primitive if it contains at least one primitive point. The bounds are derived by studying the intersections of the primitive ovoids with the configuration of those points of \(\Gamma \) which do not give rise to normal elements of E / F. Given that \(q^2+1\) is a prime number when q is even, or that \(\frac{1}{2}(q^2+1)\) is a prime number when q is odd, we actually achieve the exact number of all primitive normal elements for the quartic extension over F.

Keywords

Finite field Galois field Normal basis Primitive element Primitive normal element Projective space Ovoid Ovoid partition 

Mathematics Subject Classification

Primary 05A99 11A99 11T30 12E20 

Notes

Acknowledgements

I would like to thank Stefan Hackenberg for his kind permission to include the computational results of Table 1 in this work.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut für MathematikUniversität AugsburgAugsburgGermany

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