Separating singular moduli and the primitive element problem

We prove that ${|x-y|\ge 800X^{-4}}$, where $x$ and $y$ are distinct singular moduli of discriminants not exceeding $X$. We apply this result to the ‘primitive element problem’ for two singular moduli. In a previous article, Faye and Riffaut show that the number field ${{\mathbb{Q}}}(x,y)$, generated by two distinct singular moduli $x$ and $y$, is generated by ${x-y}$ and, with some exceptions, by ${x+y}$ as well. In this article we fix a rational number ${\alpha \ne 0,\pm 1}$ and show that the field ${{\mathbb{Q}}}(x,y)$ is generated by ${x+\alpha y}$, with a few exceptions occurring when $x$ and $y$ generate the same quadratic field over ${{\mathbb{Q}}}$. Together with the above-mentioned result of Faye and Riffaut, this generalizes a theorem due to Allombert et al. (2015) about solutions of linear equations in singular moduli.