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A New Proof of the Hansen—Mullen Irreducibility Conjecture
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- Journal:
- Canadian Journal of Mathematics / Volume 70 / Issue 6 / 01 December 2018
- Published online by Cambridge University Press:
- 20 November 2018, pp. 1373-1389
- Print publication:
- 01 December 2018
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- Article
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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.
