Is the Galois group associated to a random polynomial solvable with probability 0?

Question

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?

Answer 1

$G\cong S_n$ with probability $1$ as $n\rightarrow \infty$. This was proven first by

B. L. van der Waerden, Die Seltenheit der Gleichungen mit Affekt, Mathematische Annalen 109:1 (1934), pp. 13–16.

Look at this thread for more references.