Published online by Cambridge University Press: 05 July 2017
We classify all polynomials $P(X)\in \mathbb{Q}[X]$ with rational coefficients having the property that the quotient
$(\unicode[STIX]{x1D706}_{i}-\unicode[STIX]{x1D706}_{j})/(\unicode[STIX]{x1D706}_{k}-\unicode[STIX]{x1D706}_{\ell })$ is a rational number for all quadruples of roots
$(\unicode[STIX]{x1D706}_{i},\unicode[STIX]{x1D706}_{j},\unicode[STIX]{x1D706}_{k},\unicode[STIX]{x1D706}_{\ell })$ with
$\unicode[STIX]{x1D706}_{k}\neq \unicode[STIX]{x1D706}_{\ell }$.