Prove that both $x+y$ and $xy$ are rational, under some conditions

As a result of the answer I got for this question - Irrational solutions to some equations in two variables - I was wondering if the next statement is always true:

Let $x,y$ be real, irrational numbers such that $x+y\ne0$. And let $n_1,n_2,n_3$ be some positive integers (different from each other) such that $\gcd(n_1,n_2,n_3)=1$.

Prove (or find a counter example) that if: $$x^{n_1}+y^{n_1}$$ $$x^{n_2}+y^{n_2}$$ $$x^{n_3}+y^{n_3}$$ are all rational numbers, then also both: $$x+y$$ $$xy$$ have to be rational numbers.

Solution 1:

Sadly, the statement doesn't hold in general. For instance for $(n_1, n_2, n_3) = (2, 3, 8)$ we can choose $$ x = \frac{1}{2}\left(\sqrt{3} - 1 +\sqrt{2}\sqrt[4]{3}\right) $$ and $$ y = \frac{1}{2}\left(\sqrt{3} - 1 -\sqrt{2}\sqrt[4]{3}\right). $$ Now it's easy to see that $\gcd(2,3,8) = 1$, $x^2+y^2 = 2$, $x^3+y^3 = 2$ and $x^8+y^8 = 8$, and both numbers are irrational, but still $$ x+y = \sqrt{3} - 1 $$ and $$ xy = 1- \sqrt{3} $$ are not rational.