Question 33
Find all the functions $f : \mathbb{Q}[x] \to \mathbb{Q}[x]$ such that
i. For all $P, Q \in \mathbb{Q}[x]$ we have $f(P + Q) = f(P) + f(Q)$;
ii. For all $P \in \mathbb{Q}[x]$, we have $\gcd(P, f(P)) = 1$ if and only if $P$ is square-free.
(We call a polynomial $P \in \mathbb{Q}[x]$ square-free if there is no non-constant polynomial $Q \in \mathbb{Q}[x]$ such that $Q^2$ divides $P$.)
i. For all $P, Q \in \mathbb{Q}[x]$ we have $f(P + Q) = f(P) + f(Q)$;
ii. For all $P \in \mathbb{Q}[x]$, we have $\gcd(P, f(P)) = 1$ if and only if $P$ is square-free.
(We call a polynomial $P \in \mathbb{Q}[x]$ square-free if there is no non-constant polynomial $Q \in \mathbb{Q}[x]$ such that $Q^2$ divides $P$.)