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- Title
Separating singular moduli and the primitive element problem.
- Authors
Bilu, Yuri; Faye, Bernadette; Zhu, Huilin
- 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 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.
- Subjects
QUADRATIC fields; LINEAR equations; QUADRATIC equations; FINITE fields; RATIONAL numbers
- Publication
Quarterly Journal of Mathematics, 2020, Vol 71, Issue 4, p1253
- ISSN
0033-5606
- Publication type
Article
- DOI
10.1093/qmathj/haaa030