Factoring a binary form over a algebraically closed field

Question

Let $f(x,y)$ be a binary form over the rationals. i.e. $f\in \mathbb{Q}[x,y]$ is a homogeneous polynomial. By the Fundamental Theorem of Algebra, one has that $f$ decompose itself as a product of linear factors over $\mathbb{C}$, say $f(x,y)=\prod_{i=1}^{k}(x-\alpha_iy)$. Is there any characterization of the binary forms $f$ for which at least one of the $\alpha_i$ will be a $n$-root in $\mathbb{Q}$, i.e. there is an integer $n\geq 1$ such that $\alpha_i^n\in\mathbb{Q}$?