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International Workshop on the Arithmetic of Finite Fields

WAIFI 2016: Arithmetic of Finite Fields pp 77–83Cite as

On Sets of Irreducible Polynomials Closed by Composition

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10064))

Abstract

Let \(\mathcal {S}\) be a set of monic degree 2 polynomials over a finite field and let C be the compositional semigroup generated by \(\mathcal S\). In this paper we establish a necessary and sufficient condition for C to be consisting entirely of irreducible polynomials. The condition we deduce depends on the finite data encoded in a certain graph uniquely determined by the generating set \(\mathcal {S}\). Using this machinery we are able both to show examples of semigroups of irreducible polynomials generated by two degree 2 polynomials and to give some non-existence results for some of these sets in infinitely many prime fields satisfying certain arithmetic conditions.

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Acknowledgements

The first author was supported by the Swiss National Science Foundation grant number 168459. The second author was supported by Swiss National Science Foundation grant number 161757. The third author was supported in part by Swiss National Science Foundation grant number 149716 and Armasuisse.

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Authors and Affiliations

  1. DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilbeforce Rd, Cambridge, CB3 0WB, UK

    Andrea Ferraguti

  2. Mathematical Institute, University of Oxford, Woodstock Rd, Oxford, OX2 6GG, UK

    Giacomo Micheli

  3. Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, Zurich, 8057, Switzerland

    Reto Schnyder

Authors
  1. Andrea Ferraguti
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  2. Giacomo Micheli
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  3. Reto Schnyder
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Corresponding author

Correspondence to Giacomo Micheli .

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Editors and Affiliations

  1. University of Rennes, Rennes, France

    Sylvain Duquesne

  2. University of Leuven, Leuven, Belgium

    Svetla Petkova-Nikova

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Ferraguti, A., Micheli, G., Schnyder, R. (2016). On Sets of Irreducible Polynomials Closed by Composition. In: Duquesne, S., Petkova-Nikova, S. (eds) Arithmetic of Finite Fields. WAIFI 2016. Lecture Notes in Computer Science(), vol 10064. Springer, Cham. https://doi.org/10.1007/978-3-319-55227-9_6

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  • DOI: https://doi.org/10.1007/978-3-319-55227-9_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-55226-2

  • Online ISBN: 978-3-319-55227-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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