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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 that $G$ was a symmetric group with probability $1$ as $n\rightarrow \infty$. I will try to find a reference. – Alexander Gruber Jan 11 '13 at 18:34
up vote 22 down vote accepted

$G\cong S_n$ 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.

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For further work on this (getting error estimates on how close the probability is to 1 in a large box), look at Theorem 1.6 and the paragraph following it in – KCd Jan 11 '13 at 19:40
@KCd I cannot find the Weiss reference anymore (not even ~weiss at the technion. Any thoughts? – Igor Rivin Feb 3 '14 at 13:36
@IgorRivin: The author is Benjamin Weiss, who is at Bates now. The comment I made above was written about a year ago, and I don't know what "v56" in the URL could mean anymore, but the broken link most likely went to the paper "Probabilistic Galois of p-adic Fields", which appears now in Journal of Number Theory 133 (2013), 1537--1563. See Theorem 2.11 and the paragraph following it. – KCd Feb 3 '14 at 14:46
@KCd thanks! I am writing a survey on related matters, so this is helpful... – Igor Rivin Feb 3 '14 at 14:47
Let me put here what Weiss describes in van der Waerden's theorem and some later work. For any positive integer $B$, let $F_{n,B}$ be the set of monic, degree $n$ polynomials in $\mathbf Z[x]$ with all coefficients in $[−B + 1, B]$ and $P_n(B)$ be the proportion of polynomials in $F_{n,B}$ that are irreducible over $\mathbf Q$ and have Galois group $S_n$. Then $1− P_n(B) \ll B^{−c/\log\log B}$ as $B → \infty$, where $c = 1/(6(n-2))$. I guess $n \geq 3$. Later the error bound on $1 - P_n(B)$ was improved to $O((\log B)/\sqrt{B})$ (Gallagher) and $O_t(1/B^{t})$ for any $t < 2-\sqrt{2}$ (Dietmann). – KCd Feb 3 '14 at 14:54

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

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Thanks! Do you have a citation? – Charles Jun 23 at 18:36
@Charles Let me try and find it. I'll get back to you when I do. – Michael Tong Jun 23 at 19:11
Thanks! Mathscinet has 20 references but none of them obvious to me. – Charles Jun 23 at 19:12
@Charles I have updated the answer with an improved bound and a source for that. – Michael Tong Jun 24 at 19:08

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