On denominators of algebraic numbers and integer polynomials (Q1912291)

Let \(\alpha\) be an algebraic number with minimal polynomial \(a_dX^d+\cdots+ a_0\in[X]\). The denominator \(\text{den}(\alpha)\) of \(\alpha\) is the least positive integer \(n\) for which \(n\alpha\) is an algebraic integer. It is well known that this integer \(\text{den}(\alpha)\) divides \(a_d\). The authors compute the density of algebraic numbers \(\alpha\) of a given degree \(d\) such that \(\text{den}(\alpha)=a_d\). This density is asymptotic to \(1/\zeta(3)\) when \(d\) tends to infinity.