  Choose a polynomial $f$ uniformly at random from the set of all monic
polynomials of degree $n$ with integer coefficients in the box $[-L,L]^n$. The
main result of the paper asserts that if $L=L(n)$ grows to infinity, then the
Galois group of $f$ is the full symmetric group, asymptotically almost surely,
as $n\to \infty$.
  When $L$ grows rapidly to infinity, say $L>n^7$, this theorem follows from a
result of Gallagher. When $L$ is bounded, the analog of the theorem is open,
while the state-of-the-art is that the Galois group is large in the sense that
it contains the alternating group (if $L< 17$, it is conditional on the general
Riemann hypothesis). Hence the most interesting case of the theorem is when $L$
grows slowly to infinity.
  Our method works for more general independent coefficients.
