Full Galois groups of polynomials with slowly growing coefficients

Choose a polynomial (Formula presented.) uniformly at random from the set of all monic polynomials of degree (Formula presented.) with integer coefficients in the box (Formula presented.). The main result of the paper asserts that if (Formula presented.) grows to infinity, then the Galois group of (Formula presented.) is the full symmetric group, asymptotically almost surely, as (Formula presented.). When (Formula presented.) grows rapidly to infinity, say (Formula presented.), this theorem follows from a result of Gallagher. When (Formula presented.) 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 (Formula presented.), it is conditional on the Extended Riemann Hypothesis). Hence the most interesting case of the theorem is when (Formula presented.) grows slowly to infinity. Our method works for more general independent coefficients.