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arXiv · 1910.06388

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

Matthew Friedrichsen·Daniel Keliher
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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.

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Matthew Friedrichsen, Daniel Keliher. 2019-10-14. Comparing the density of D_4 and S_4 quartic extensions of number fields. https://arxiv.org/abs/1910.06388

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