Density of polynomials with a prescribed number field extension

Question

For any polynomial $f(x) \in \mathbb{Z}[x]$, let $K_f$ denote the minimum splitting field over $\mathbb{Q}$ which contains all of the roots of $f$. Let $n \geq 2$ be a fixed integer, and let $K$ be a fixed number field which is the splitting field of some polynomial $f(x) \in \mathbb{Z}[x]$ of degree $n$. Let

$$\displaystyle \mathcal{S}_N = \{f(x) = a_n x^n + \cdots + a_1 x + a_0 \in \mathbb{Z}[x]: |a_i| \leq N, i = 0, \cdots, n, K_f = K \}.$$

Then is there an estimate for the quantity

$$\displaystyle \# \mathcal{S}_N/N^{n+1}?$$

More precisely, can we obtain an exact asymptotic formula, in terms of $N$ and $K$, for $\# \mathcal{S}_N$?

Answer 1

This sort of question goes back to van der Waerden (who showed that "most" polynomials have the full symmetric group as the Galois group). The most recent (and most relevant) result of this sort are due to Dietmann, and in Dietmann, Rainer(4-LNDHB) On the distribution of Galois groups. (English summary) Mathematika 58 (2012), no. 1, 35–44.

he showed that the number of monic polynomials of height $H$ (your $N$) with fixed Galois group $G$ is bounded as $$\ll_{n, \epsilon} H^{n-1 +\frac{|G|}{n!} + \epsilon}.$$ Your question is not quite the same, but close enough.