Multiplication of two irrational to give rational

Question

I know that we can obtain any rational $r\in\mathbb{R}\setminus\{0\}$ by the multiplication of two irrational numbers. There are many beautiful answers to this here.

But I want to know that if there exists a theorem/result which can exactly point out that when does (a shot of classification that) the multiplication of given two irrational numbers is a rational number?

If the above thing is much more to ask for, then so can we expect that we have a finite list of the product of two irrational numbers that are unsettled and others flow some general pattern or we have some theorems?

Answer 1

You ask:

I want to know that if there exists a theorem/result which can exactly point out that when does (a shot of classification that) the multiplication of given two irrational numbers is a rational number?

Ummm... just multiply them and find out. Works most of the time!