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

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.

Enumerating $D_4$ Quartics and a Galois Group Bias Over Function Fields

“…Finally, note that taking q → ∞ and large g in Theorem 1.4 is essentially an extremal version of [FK,Corollary 1.2] but for function fields rather than number fields. It is plausible that the L-function techniques used in the number field version could be ported over to the function field setting in order to ease the condition on q.…”