Is there a non trivial set of pairs of irrationals that give a rational when multiplied

Question

I'm wondering if there exists a non trivial set of pairs of irrationals that gives a rational when multiplied together. What I mean with trivial is for instance the set of pairs composed of an irrational and one of its multiplicative inverse : $\{(x,r/x) \ | \ x \in \mathbb{R} \setminus \mathbb{Q} \text{ and } r \in \mathbb{Q}^*\}$.

For instance let $K_{\sqrt{3}}=\{x \in \mathbb{R} \setminus \mathbb{Q} \ | \ x\sqrt{3} \in \mathbb{Q} \} $. We know that the rational multiplicatives of $\sqrt{3}$ belong to $K_{\sqrt{3}}$. Is there anything else ?

Answer 1

Of course any pair of irrational numbers whose product is rational is necessarily of the trivial form you mention.

That's just because if $xy=r\in\mathbb Q$, then $y=r/x$ and the pair is $(x,r/x)$ (we assume $r\neq 0$).

Answer 2

If $a$ and $b$ are irrational yet $ab=q$ is rational then $a=\frac{q}{b}$ so your pair is $b$ and $\frac{q}{b}$ which is trivial by your definition.

(Of course, I can divide by $b$ since $0$ isn't irrational.)