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

Probabilistic Galois Theory -- The Square Discriminant Case