# Coprimality at compositions of rational expressions in general?

Let $$P_1(X),P_2(X),Q_1(X),Q_2(X)\in\overline{\mathbb{Q}}[X]$$ (or $$\mathbb{Q}[X]$$) and $$r(X)\in\overline{\mathbb{Q}}(X)$$ (or $$\mathbb{Q}(X)$$).
Let $$\frac{P_1(r(x))}{Q_1(r(x))}=\frac{P_2(x)}{Q_2(x)}$$.

Let's consider coprimality over $$\overline{\mathbb{Q}}$$ (or $$\mathbb{Q}$$).

1.)
In which cases can the coprimality of $$P_2(x)$$ and $$Q_2(x)$$ be concluded from the coprimality of $$P_1(x)$$ and $$Q_1(x)$$?

2.)
In which cases can the coprimality of $$P_1(x)$$ and $$Q_1(x)$$ be concluded from the coprimality of $$P_2(x)$$ and $$Q_2(x)$$?

3.) Is this possible at least for some kinds of $$r(X\in\overline{\mathbb{Q}}[X]$$ (or $$\mathbb{Q}[X]$$)?

I don't know how I could start to derive an answer.

The question is important for answering the question when an irreducible rational equation keeps irreducible after inserting a rational expression.

I need the answer because I want to try to prove some of the conjectures in Closed-form solubility of elementary transcendental equations?.

Both answers are "always", at least when $$r$$ is nonconstant.
(1): If $$P_2(x)$$ and $$Q_2(x)$$ have the factor $$G(x)$$ in common, then let $$\alpha$$ be a root of $$G(x)$$ in $$\overline{\Bbb Q}$$. We have $$P_2(\alpha)=Q_2(\alpha)=0$$, and so if $$\beta=r(\alpha)$$, we have $$P_1(\beta)=Q_1(\beta)=0$$. Then if $$H(x)$$ is the minimal polynomial of $$\beta$$ (over whatever field we're working over), we conclude that $$P_1(x)$$ and $$Q_1(x)$$ have the factor $$H(x)$$ in common.
(2): If $$P_1(x)$$ and $$Q_1(x)$$ have the factor $$F(x)$$ in common, then $$P_2(x)$$ and $$Q_2(x)$$ will have the factor $$F(r(x))$$ in common.