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

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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