Is there a set the subproducts of which are always rational.

Question

This question is inspired from the answer of: Is every subproduct of rational product is rational?

Is there a set $P$ for which: $$\prod_{p \in P}a_p=I$$ while $$\prod_{p \in P_1}a_p=R$$ for every $P_1 \subset P$, where $R$ is rational, $I$ is irrational, $a_p$ is rational and $0<a_p \le 1$.

Answer 1

If $a_p$ is rational for all $p \in P$, and if the product

$$\prod_{p \in P}a_p$$

is irrational, then for any $p_1 \in P$, the set $P_1 = P - \{p_1\}$ is such that the product

$$\prod_{p \in P_1}a_p$$

is also irrational.