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On the first page of his paper where he proves van der Waerden’s conjecture, Bhargava mentions that Hilbert’s irreducibility theorem shows that the number of monic integer polynomials of degree $n$, all of whose coefficients have absolute value less than $H$, which have Galois group not equal to $S_n$ is $o(H^n)$. I suppose the argument he’s referring to is well-known or straightforward, although he does not cite a source. What is the argument?

Vik78
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The argument is old, going back to van der Waerden, with improvements by later authors such as Gallagher. See here (look for the notation $E_n(N)$) and Theorem 1.2 here.

KCd
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  • Excellent, thank you. I will accept the answer once I have a chance to read the sources you linked. – Vik78 May 23 '22 at 01:21
  • The paper of van der Waerden is mentioned on the MSE page https://math.stackexchange.com/questions/2071358/making-precise-generally-the-galois-group-is-s-n. Note the German terms “mit Affekte” (with Affect) and “ohne Affekt” (without Affect) refer to a degree-$n$ equation having Galois group $S_n$ or Galois group smaller than $S_n$. I do not know what “Affekt” is really meaning literally here (why being “with it” means all permutation appear in the Galois group). – KCd May 23 '22 at 01:38
  • @KCd I think you've got it backwards: "ohne Affekt" means Galois group $S_n$ and "mit Affekt" means smaller Galois group. I also have no idea where this terminology comes from. – Lukas Heger May 23 '22 at 14:50
  • @Lukas Heger yes, I agree you are probably right. It is hard for me to remember what those two terms mean (like “inessential discriminant divisor” in algebraic number theory). – KCd May 23 '22 at 15:20