The van der Waerden's Conjecture states that the set $\mathscr{ P}_{ n,N}^0(\mathbb{ Q})$ of monic integer polynomials $f(X)$ of degree $n$, with height $\le N$ such that the Galois group $G_{ K_f/\mathbb{ Q}}$ of the splitting field $K_f/\mathbb{ Q}$ is the full symmetric group, has order $|\mathscr{ P}_{ n,N}^0(\mathbb{ Q})|=(2N)^n+O_n(N^{ n-1})$ as $N\rightarrow+\infty$. The conjecture has been shown previously for cubic and quartics polynomials by van der Waerden, Chow and Dietmann. Subsequently, Bhargava proved it for $n\ge6$. 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 $K/\mathbb{ Q}$ of degree $d$, for some values of $n$ and $d$.
Categories: math.NT