An Average Version of the Chebotarev Density Theorem for Families of Galois Extensions

We discuss the arithmetic of certain families of number fields, obtained as the splitting fields of families of polynomials. The first and main example is the family of monic, integer polynomials over a fixed number field extension of fixed degree, with bounded height. It was van der Waerden who first proved that almost all (in a suitable sense) such polynomials have splitting field over the rationals with Galois group isomorphic to the symmetric group. On the other hand, all these kinds of number field extensions arise in this way for some polynomial as above. We show an average version of the Chebotarev Density Theorem for this family of polynomials. In particular, this gives a Central Limit Theorem for the number of primes with given splitting type in some ranges.