The way I understand it, is that if $f(x)$ is an irreducible polynomial in $\mathbb{Q}[x]$ of degree at least 2, then a difference of distinct roots $a_i-a_j$ is never rational for any of the $a_1,\dots,a_n$ which are the roots of $f(x)$ in $\mathbb{C}$.
Why is this? If $a_i-a_j\in\mathbb{Q}$ for some distinct roots, what goes wrong? Would it follow somehow that $f(x)$ is reducible over $\mathbb{Q}$? Or perhaps there's a more direct explanation?