On perfect polynomials over $\mathbb{F}_p$ with $p$ irreducible factors

Abstract

We consider, for a fixed odd prime number p , monic polynomials in one variable over the finite field \mathbb{F}_p which are equal to the sum of their monic divisors. Call them \emph{perfect} polynomials. We prove that the exponents of each irreducible factor of any perfect polynomial having no root in \mathbb{F}_p and p irreducible factors are all less than p-1 . We completely characterize those perfect polynomials for which each irreducible factor has degree two and all exponents do not exceed two.