PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Representation of a polynomial as the sum of an irreducible polynomial and a square-free polynomial

Volume 197 / 2021

Mireille Car, Luis H. Gallardo Acta Arithmetica 197 (2021), 293-309 MSC: Primary 11T55; Secondary 11T06. DOI: 10.4064/aa200324-30-7 Published online: 15 December 2020


Let ${\mathbb F }$ be a finite field with $q$ elements. We prove that every polynomial $M\in {\mathbb F }[T]$ of degree large enough is a sum $P+Q$ where $P$ is an irreducible polynomial with $\deg P=\deg M$ and $Q$ is a square-free polynomial with $\deg Q\leq \deg M$. Together with some computer calculations we extend that result to all non-constant polynomials $M\in {\mathbb F }[T]$ for $q \gt 2$ and to all non-constant polynomials of degree other than $2$, $30$, $31$, $32$, for $q=2$. For $q=2$, polynomials of degree $30$, $31$ and $32$ remain unchecked.


  • Mireille CarAix-Marseille Université
    Institut de Mathématiques de Marseille
    CNRS, UMR 7373
    CMI, 39 rue F. Joliot-Curie
    F-13453 Marseille Cedex 13, France
  • Luis H. GallardoUniv. Brest
    UMR CNRS 6205
    Laboratoire de Mathématiques
    de Bretagne Atlantique
    6, Av. Le Gorgeu
    C.S. 93837
    F-29238 Brest Cedex 3, France

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image