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
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 (which produce the coefficients
of characteristic polynomials)
satisfies a simple requirement on the least period.
This bears a sharp contrast to previous techniques in literature
employed to tackle existence
of irreducible polynomials with prescribed coefficients.
irreducible polynomial, primitive polynomial, Hansen-Mullen conjecture, symmetric function, $q$-symmetric, discrete Fourier transform, finite field
11T06 - Polynomials