Galois Groups of Generic Polynomials

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We show that the Galois group of a random monic polynomial %of degree d>12 with integer coefficients between -N and N is NOT Sd with probability ≪ Ω⁽ᵈ⁾NN. Conditionally on NOTbeing the full symmetric group, we have a hierarchy of possibilities each of which has polylog probability of occurring. These results also apply to random polynomials with only a subset of the coefficients allowed to vary. This settles a question going back to 1936.