Density of Special Classes of Polynomials with Squarefree Discriminant

In this paper, we consider the problem of determining the density of monic polynomials over ${\mathbb{Z}}_p$ with squarefree discriminant over various subsets of the set of monic polynomials over ${\mathbb{Z}}_p$ of fixed degree. We compute the density of polynomials in each subset whose discriminant is squarefree, and we compute the density of polynomials $f$ in each subset such that ${\mathbb{Z}}_p[x]\mathrm{/}(f(x))$ is the maximal order of ${\mathbb{Q}}_p[x]\mathrm{/}(f(x))$.