Mass Formulae for Extensions of Local Fields, and Conjectures on the Density of Number Field Discriminants

Using Serre's mass formula 15 for totally ramified extensions, we derive a mass formula that counts all (isomorphism classes of) étale algebra extensions of a local field F having a given degree n. Along the way, we also prove a series of mass formulae for counting étale extensions of a local field F having certain properties, such as a chosen prime splitting or ramification behavior. We then use these mass formulae to formulate a heuristic that predicts the asymptotic number of number fields, of fixed degree n and bounded discriminant, whose Galois closures have Galois group the full symmetric group Sn over ℚ. Analogous predictions are made for the asymptotic density of those number fields having specified local behaviors at finitely many places. All these predictions are in full agreement with the known results in degrees up to five.