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

Friedrichsen, Matthew; Keliher, Daniel

doi:10.1090/proc/15358

Comparing the density of $D_4$ and $S_4$ quartic extensions of number fields
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by Matthew Friedrichsen and Daniel Keliher PDF

Proc. Amer. Math. Soc. 149 (2021), 2357-2369

Abstract:

When ordered by discriminant, it is known that about 83% of quartic fields over $\mathbb {Q}$ 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 Generalized Riemann Hypothesis, we give a lower bound that holds for general number fields.

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