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Proceedings of the American Mathematical Society

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


Authors: Matthew Friedrichsen and Daniel Keliher
Journal: Proc. Amer. Math. Soc. 149 (2021), 2357-2369
MSC (2020): Primary 11R42, 11R29, 11R45, 11R16
DOI: https://doi.org/10.1090/proc/15358
Published electronically: March 23, 2021
MathSciNet review: 4246788
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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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Additional Information

Matthew Friedrichsen
Affiliation: Department of Mathematics, Tufts University, Medford, Massachusetts 02155
MR Author ID: 1273228
ORCID: 0000-0002-9564-0451
Email: matthew.friedrichsen@tufts.edu

Daniel Keliher
Affiliation: Department of Mathematics, Tufts University, Medford, Massachusetts 02155
ORCID: 0000-0001-5144-7898
Email: daniel.keliher@tufts.edu

Received by editor(s): October 22, 2019
Received by editor(s) in revised form: September 28, 2020
Published electronically: March 23, 2021
Communicated by: Amanda Folsom
Article copyright: © Copyright 2021 Copyright by the authors