# Statistic of cubic Irreducible polynomial with cyclic Galois group

Van der Waerden proved for monic irreducible polynomials $f$ of degree $n$ with bounded height $X$:
$$|\{ f(x)\in \mathbb{Z}[x]: Gal(f)\neq S_n \}|=o(X)$$ Where height of $f$ is defined by maximum of absolute value of its coefficient.

So if we order monic irreducible polynomial of degree $n$ with respect to height, then we expect to obtain symmetric group as its Galois group. Now what if we want to replace height by discriminant.

$$|\{ f\in \mathbb{Z}[x]: monic, irreducible \deg(f)=n, |Disc(f)|\leq X, Gal(f)\neq S_n\}|=o(X)$$
@Franz: I am a bit familiar with Bhargava results, but here I want to count polynomials instead to count number field, I think this would be not exactly similar to counting number field, since different polynomials can give same number field with the same discriminant. Also for example if one look at quartic number field then contribution of $S_4$ and $D_4$ are linear i.e cx. I expect for polynomials contribution of $D_4$ is negligible. – M.B Nov 16 '10 at 23:11