TITLE: Is the Galois group associated to a random polynomial solvable with probability 0?
QUESTION [39 upvotes]: Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$. 
Let $r_1,\ldots,r_n$ be the roots of $P$ and consider $$G=\operatorname{Gal}(\mathbb{Q}(r_1,\ldots,r_n)/\mathbb{Q})$$
What is the probability, as $n\to\infty,$ that $G$ is solvable? (I assume 0.) Who first proved this?

REPLY [2 votes]: Yes, the probability is zero since it equals the symmetric group with probability one:
If $Q_d(N)$ denotes the set of degree $d$ polynomials with coefficients $|a_i|\le N$ with Galois group not equal to $S_d$, then
$$
|Q_d(N)|\ll N^{d-1/2}\log N
$$
This bound is sufficient to prove your result. This was proven by Patrick X. Gallagher.
EDIT: Gallagher proved the following stronger result:
$$|Q_d(N)| \ll N^{d-1/2} \log^{1 - \gamma} N$$
where $\gamma \sim (2 \pi d)^{-1/2}$. Source