Probabilistic Galois Theory: The square discriminant case

Abstract The paper studies the probability for a Galois group of a random polynomial to be . We focus on the so‐called large box model, where we choose the coefficients of the polynomial independently and uniformly from . The state‐of‐the‐art upper bound is , due to Bhargava. We conjecture a much stronger upper bound , 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.