  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.
