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Designs, Codes and Cryptography

, Volume 87, Issue 7, pp 1657–1671 | Cite as

Factorization of a class of composed polynomials

  • Lucas ReisEmail author
Article
  • 110 Downloads

Abstract

In this paper, we provide the degree distribution of irreducible factors of the composed polynomial f(L(x)) over \(\mathbb {F}_q\), where \(f(x)\in \mathbb {F}_q[x]\) is irreducible and \(L(x)\in \mathbb {F}_q[x]\) is a linearized polynomial. We further provide some applications of our main result, including lower bounds for the number of irreducible factors of f(L(x)), constructions of high degree irreducible polynomials and the explicit factorization of \(f(x^q-x)\) under certain conditions on f(x).

Keywords

Factorization Finite fields Linearized polynomials \(\mathbb {F}_q\)-order 

Mathematics Subject Classification

12E20 11T30 

Notes

Acknowledgements

This work was conducted during a visit to Carleton University, supported by the program CAPES-PDSE (process - 88881.134747/2016-01).

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal de Minas GeraisBelo HorizonteBrazil
  2. 2.School of Mathematics and StatisticsCarleton UniversityOttawaCanada

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