The density of discriminants of quintic rings and fields

TL;DR

This paper establishes the asymptotic count of quintic fields with bounded discriminant, showing that almost all such fields have Galois group S_5, and interprets related constants via local mass formulas.

Contribution

It provides the first asymptotic enumeration of quintic fields by discriminant and demonstrates that 100% have Galois group S_5, extending understanding beyond quartic fields.

Findings

Asymptotic count of quintic fields with discriminant ≤ X is proportional to X.

Almost all quintic fields have Galois closure with Galois group S_5.

Constants in the theorems are interpreted through local mass calculations.

Abstract

We determine, asymptotically, the number of quintic fields having bounded discriminant. Specifically, we prove that the asymptotic number of quintic fields having absolute discriminant at most X is a constant times X. In contrast with the quartic case, we also show that a density of 100% of quintic fields, when ordered by absolute discriminant, have Galois closure with full Galois group $S_5$. The analogues of these results are also proven for orders in quintic fields. Finally, we give an interpretation of the various constants appearing in these theorems in terms of local masses of quintic rings and fields.