Igor Rivin
Published 2015-11-19Version 1
We show that the Galois group of a random monic polynomial %of degree $d>12$ with integer coefficients between $-N$ and $N$ is NOT $S_d$ with probability $\ll \frac{\log^{\Omega(d)}N}{N}.$ Conditionally on NOTbeing the full symmetric group, we have a hierarchy of possibilities each of which has polylog probability of occurring. These results also apply to random polynomials with only a subset of the coefficients allowed to vary. This settles a question going back to 1936.
On the distribution of Galois groups
Galois groups of $\binom{n}{0} + \binom{n}{1} X + \ldots + \binom{n}{6} X^6$
Polynomials with roots in ${\Bbb Q}_p$ for all $p$
![QR code for arXiv:1511.06446 [math.NT]](http://oss.arxitics.com/images/favicon.svg)