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?

  • 2
    $\begingroup$ I remember reading that $G$ was a symmetric group with probability $1$ as $n\rightarrow \infty$. I will try to find a reference. $\endgroup$
    – Alexander Gruber
    Jan 11, 2013 at 18:34

2 Answers 2


$G\cong S_n$ 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.

  • 3
    $\begingroup$ 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 www.technion.ac.il/~weiss/Distn-v56.pdf. $\endgroup$
    – KCd
    Jan 11, 2013 at 19:40
  • $\begingroup$ @KCd I cannot find the Weiss reference anymore (not even ~weiss at the technion. Any thoughts? $\endgroup$
    – Igor Rivin
    Feb 3, 2014 at 13:36
  • $\begingroup$ @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. $\endgroup$
    – KCd
    Feb 3, 2014 at 14:46
  • $\begingroup$ @KCd thanks! I am writing a survey on related matters, so this is helpful... $\endgroup$
    – Igor Rivin
    Feb 3, 2014 at 14:47
  • 2
    $\begingroup$ 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). $\endgroup$
    – KCd
    Feb 3, 2014 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

  • $\begingroup$ Thanks! Do you have a citation? $\endgroup$
    – Charles
    Jun 23, 2016 at 18:36
  • $\begingroup$ @Charles Let me try and find it. I'll get back to you when I do. $\endgroup$
    – MCT
    Jun 23, 2016 at 19:11
  • $\begingroup$ Thanks! Mathscinet has 20 references but none of them obvious to me. $\endgroup$
    – Charles
    Jun 23, 2016 at 19:12
  • 1
    $\begingroup$ @Charles I have updated the answer with an improved bound and a source for that. $\endgroup$
    – MCT
    Jun 24, 2016 at 19:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.