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A function on Q[x]

Source: Iran Team selection test 2024 - P8

May 19, 2024
algebra

Problem Statement

Find all functions f:Q[x]→Q[x]f : \mathbb{Q}[x] \to \mathbb{Q}[x] such that two following conditions holds : ∀P,Q∈Q[x]:f(P+Q)=f(P)+f(Q)\forall P , Q \in \mathbb{Q}[x] : f(P+Q)=f(P)+f(Q) ∀P∈Q[x]:gcd(P,f(P))=1  ⟺  \forall P \in \mathbb{Q}[x] : gcd(P , f(P))=1 \iff PP 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
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