A New Proof of the Hansen—Mullen Irreducibility Conjecture

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 (DFT) of functions with values in finite fields. We exploit this relation and prove, in an elementary fashion, that a relevant function related to the 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.