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

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?

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I remember reading the result that $G$ is a symmetric group with probability $1$ as $n\rightarrow \infty$. I will try to find a reference. –  Alexander Gruber Jan 11 at 18:34

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