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

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- by Matthew Friedrichsen and Daniel Keliher
- Proc. Amer. Math. Soc.
**149**(2021), 2357-2369 - DOI: https://doi.org/10.1090/proc/15358
- Published electronically: March 23, 2021

## 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.## References

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## Bibliographic 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
- © Copyright 2021 Copyright by the authors
- Journal: Proc. Amer. Math. Soc.
**149**(2021), 2357-2369 - MSC (2020): Primary 11R42, 11R29, 11R45, 11R16
- DOI: https://doi.org/10.1090/proc/15358
- MathSciNet review: 4246788