Comparing the density of D4 and S4 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 S4, while the remaining 17% have Galois group D4. We study these proportions over a general number field F. We find that asymptotically 100% of quadratic number fields have more D4 extensions than S4 and that the ratio between the number of D4 and S4 quartic extensions is biased arbitrarily in favor of D4 extensions. Under GRH, we give a lower bound that holds for general number fields.
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