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A New Proof of the Hansen—Mullen Irreducibility Conjecture

Published online by Cambridge University Press:  20 November 2018

Aleksandr Tuxanidy  and
Qiang Wang
Aleksandr Tuxanidy
Affiliation:
School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, e-mail: aleksandrtuxanidytor@cmail.carleton.ca , wang@math.carleton.ca
Qiang Wang
Affiliation:
School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, e-mail: aleksandrtuxanidytor@cmail.carleton.ca , wang@math.carleton.ca
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Abstract

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We give a new proof of the Hansen–Mullen irreducibility conjecture. The proof relies on an application of a (seemingly new) sufficient condition for the existence of elements of degree $n$ in the support of functions on finite fields. This connection to irreducible polynomials is made via the least period of the discrete Fourier transform $\left( \text{DFT} \right)$ of functions with values in finite fields. We exploit this relation and prove, in an elementary fashion, that a relevant function related to the $\text{DFT}$ of characteristic elementary symmetric functions (that produce the coefficients of characteristic polynomials) satisfies a simple requirement on the least period. This bears a sharp contrast to previous techniques employed in the literature to tackle the existence of irreducible polynomials with prescribed coefficients.

Keywords

11T06irreducible polynomialprimitive polynomialHansen–Mullen conjecturesymmetric functionq-symmetricdiscrete Fourier transformfinite field
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 6 , 01 December 2018 , pp. 1373 - 1389
Copyright
Copyright © Canadian Mathematical Society 2018

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