Let $a$ and $b$ be positive rational numbers and $n\ge 2$ be an integer so that $\sqrt[n]{a} + \sqrt[n]{b} \in \mathbb{Q}$. Prove that $\sqrt[n]{a}\in \mathbb{Q}$.
We can obviously write $a=p_1/q_1, b = p_2/q_2$ for coprime positive integers $p_1,q_1$ and $p_2,q_2$. Let $c=\sqrt[n]{a} + \sqrt[n]{b}$. It could be useful to find a common root of some polynomials, one of which is $x^n-a$. I'm not sure what other polynomials to consider (e.g. $x^n-b, (c-x)^n - b, (c-x)^n - a$). Note that both $x^n-b$ and $(c-x)^n-a$ share a common root in $\mathbb{C}$, but I'm not sure whether this root is unique. The uniqueness of the root could be useful for proving that $\sqrt[n]{b}$ is rational, which is equivalent to the problem statement.
