A positive proportion of quartic fields are not monogenic yet have no local obstruction to being so

Levent Alpöge; Manjul Bhargava; Ari Shnidman

doi:10.1007/s00208-023-02575-0

A positive proportion of quartic fields are not monogenic yet have no local obstruction to being so

Levent Alpöge, Manjul Bhargava, Ari Shnidman^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We show that a positive proportion of quartic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof builds on the corresponding result for cubic fields that we obtained in a previous work. Along the way, we also prove that a positive proportion of quartic rings of integers do not arise as the invariant order of an integral binary quartic form despite having no local obstruction.

Original language	American English
Journal	Mathematische Annalen
DOIs	https://doi.org/10.1007/s00208-023-02575-0
State	Accepted/In press - 2023

Bibliographical note

Funding Information:
The authors thank Dan Fess and Arul Shankar for many helpful conversations. The first author was supported by NSF Grant DMS-2002109. The second author was supported by a Simons Investigator Grant and NSF Grant DMS-1001828. The third author was supported by the Israel Science Foundation (Grant No. 2301/20).

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

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10.1007/s00208-023-02575-0

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A positive proportion of quartic fields are not monogenic yet have no local obstruction to being so

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