Full Galois groups of polynomials with slowly growing coefficients

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