JournalspmVol. 69 / No. 4DOI 10.4171/pm/1918

On perfect polynomials over Fp\mathbb{F}_p with pp irreducible factors

  • Luis H. Gallardo

    Université de Brest, France
  • Olivier Rahavandrainy

    Université de Brest, France
On perfect polynomials over $\mathbb{F}_p$ with $p$ irreducible factors cover

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Abstract

We consider, for a fixed odd prime number , monic polynomials in one variable over the finite field 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 and irreducible factors are all less than . We completely characterize those perfect polynomials for which each irreducible factor has degree two and all exponents do not exceed two.