The density of discriminants of S₃-sextic number fields

We prove an asymptotic formula for the number of sextic number fields with Galois group S₃ and absolute discriminant <X. In addition, we give an interpretation of the constant in the formula in terms of the asymptotic densities of given local completions among these sextic fields. Our proof gives analogous results when we count S₃-sextic extensions of any number field, and also when finitely many local completions have been specified for the sextic extensions.