For fixed $n> 1$, consider the space of degree-$n$ monic polynomials. Lenstra conjuectured that the density of polynomials f(x) such that $f$ is irreducible and $\mathbb{Z}[x]/f(x)$ is the ring of integers in its fraction field is $\zeta(2)^{-1}$. Similar conjectures have been made for the density of degree-$n$ polynomials with squarefree discriminants. In Lenstra's conjecture, the polynomials are ordered by the size of their largest coefficient. We order these polynomials by a different height function, namely, the size of their largest root. We will show that Lenstra's conjecture is true under this new ordering.

This is joint work with Manjul Bhargava and Xiaoheng Wang.