Representation of a polynomial as the sum of an irreducible polynomial and a square-free polynomial
Volume 197 / 2021
Acta Arithmetica 197 (2021), 293-309
MSC: Primary 11T55; Secondary 11T06.
DOI: 10.4064/aa200324-30-7
Published online: 15 December 2020
Abstract
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.
