  The paper studies the probability for a Galois group of a random polynomial
to be $A_n$. We focus on the so-called large box model, where we choose the
coefficients of the polynomial independently and uniformly from $\{-L,\ldots,
L\}$. The state-of-the-art upper bound is $O(L^{-1})$, due to Bhargava. We
conjecture a much stronger upper bound $L^{-n/2 +\epsilon}$, and that this
bound is essentially sharp. We prove strong lower bounds both on this
probability and on the related probability of the discriminant being a square.
