# Separating singular moduli and the primitive element problem

## TL;DR

This paper establishes a lower bound on the difference between distinct singular moduli and applies it to determine generators of certain number fields formed by two singular moduli, significantly extending previous results.

## Contribution

It proves a new inequality for singular moduli differences and demonstrates that fields generated by two singular moduli can be generated by linear combinations involving a fixed rational number.

## Key findings

- Established a lower bound |x-y| ≥ 800X^{-4} for distinct singular moduli
- Proved that Q(x,y) is generated by x + α y for fixed rational α ≠ 0, ±1
- Extended previous theorems on solutions of linear equations in singular moduli

## Abstract

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 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 $\alpha \ne0,\pm1$ 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 gives a drastic generalization of a theorem due to Allombert et al. (2015) about solution of linear equations in singular moduli.