JournalspmVol. 69, No. 4pp. 283–303

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 pp, monic polynomials in one variable over the finite field Fp\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 Fp\mathbb{F}_p and pp irreducible factors are all less than p1p-1. We completely characterize those perfect polynomials for which each irreducible factor has degree two and all exponents do not exceed two.

Cite this article

Luis H. Gallardo, Olivier Rahavandrainy, On perfect polynomials over Fp\mathbb{F}_p with pp irreducible factors. Port. Math. 69 (2012), no. 4, pp. 283–303

DOI 10.4171/PM/1918