# How to prove that polynomials with integer coefficients generically have full Galois group

Based on the graphic in the MathWorld article on the quintic equation it seems very likely that following statement is either true or trivially false in a way that can be easily remedied by adding an additional hypothesis:

Let $A_N$ be the set of degree $n$ polynomials $f(x)$ with integer coefficients $a_0, a_1, ... a_n$ such that $|a_i| < N$. Let $S_N$ be the fraction of these that have Galois group $S_n$. Then as $N \rightarrow \infty$ , $S_N \rightarrow 1$.

It also seems likely that this result follows from well known results: how does one prove it?

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Thoroughly discussed on MO: mathoverflow.net/questions/58397/… –  Qiaochu Yuan Jan 3 '12 at 6:15