Comparing the density of D_4 and S_4 quartic extensions of number fields
Number Theory
2021-05-13 v2
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.
Cite
@article{arxiv.1910.06388,
title = {Comparing the density of D_4 and S_4 quartic extensions of number fields},
author = {Matthew Friedrichsen and Daniel Keliher},
journal= {arXiv preprint arXiv:1910.06388},
year = {2021}
}
Comments
Fixed a typo with Theorem 1.3 that is present in the published version of the paper. The main results remain unchanged