Problem

Source: Iran Team selection test 2024 - P8

Tags: algebra



Find all functions $f : \mathbb{Q}[x] \to \mathbb{Q}[x]$ such that two following conditions holds : $$\forall P , Q \in \mathbb{Q}[x] : f(P+Q)=f(P)+f(Q)$$$$\forall P \in \mathbb{Q}[x] : gcd(P , f(P))=1 \iff$$$P$ is square-free. Which a square-free polynomial with rational coefficients is a polynomial such that there doesn't exist square of a non-constant polynomial with rational coefficients that divides it. Proposed by Sina Azizedin