Towards a generalization of the van der Waerden's conjecture for Sn-polynomials with integral coefficients over a fixed number field extension

The van der Waerden’s Conjecture states that the set ${\mathcal{P}}_{n,N}^0(\mathbb{Q})$ of monic integer polynomials $f(X)$ of degree $n$, with height $\le N$ such that the Galois group $G_{K_f\mathrm{/}\mathbb{Q}}$ of the splitting field $K_f\mathrm{/}\mathbb{Q}$ is the full symmetric group, has order $\mathrm{\mid }{\mathcal{P}}_{n,N}^0(\mathbb{Q})\mathrm{\mid }=(2N{)}^n+O_n(N^{n-1})$ as $N\to +\mathrm{\infty }$. The conjecture has been shown previously for cubic and quartics polynomials by van der Waerden, Chow and Dietmann. Subsequently, Bhargava proved it for $n\ge 6$. In this paper, we generalize the result for polynomials with coefficients in the ring of algebraic integers ${\mathcal{O}}_K$ of a fixed finite extension $K\mathrm{/}\mathbb{Q}$ of degree $d$, for some values of $n$ and $d$.