Comparing the density of D_4 and S_4 quartic extensions of number fields

TL;DR

This paper investigates the distribution of D_4 and S_4 quartic extensions over various number fields, revealing a bias towards D_4 extensions and providing bounds under GRH.

Contribution

It extends known results from Q to general number fields, showing a dominant presence of D_4 extensions and analyzing their ratio bias.

Findings

Over quadratic fields, D_4 extensions are more common than S_4.

The ratio of D_4 to S_4 extensions can be arbitrarily biased.

Under GRH, a lower bound for the ratio is established.

Abstract

When ordered by discriminant, it is known that about 83% of quartic fields over Q have associated Galois group S_4, while the remaining 17% have Galois group D_4. We study these proportions over a general number field F. We find that asymptotically 100% of quadratic number fields have more D_4 extensions than S_4 and that the ratio between the number of D_4 and S_4 quartic extensions is biased arbitrarily in favor of D_4 extensions. Under GRH, we give a lower bound that holds for general number fields.