Comparing the density of 𝐷₄ and 𝑆₄ quartic extensions of number fields

When ordered by discriminant, it is known that about 83% of quartic fields over Q \mathbb {Q} have associated Galois group S 4 S_4 , while the remaining 17% have Galois group D 4 D_4 . We study these proportions over a general number field F F . We find that asymptotically 100% of quadratic number fields have more D 4 D_4 extensions than S 4 S_4 and that the ratio between the number of D 4 D_4 and S 4 S_4 quartic extensions is biased arbitrarily in favor of D 4 D_4 extensions. Under Generalized Riemann Hypothesis, we give a lower bound that holds for general number fields.