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

Mathematics

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

2 Scopus citations

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	English (US)
Pages (from-to)	4037-4052
Number of pages	16
Journal	Mathematische Annalen
Volume	388
Issue number	4
DOIs	https://doi.org/10.1007/s00208-023-02575-0
State	Published - Apr 2024

All Science Journal Classification (ASJC) codes

General Mathematics

Access to Document

10.1007/s00208-023-02575-0

Cite this

@article{d4edd62ffd804707ad051c4cb778f4e2,

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

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.",

author = "Levent Alp{\"o}ge and Manjul Bhargava and Ari Shnidman",

note = "Publisher Copyright: {\textcopyright} The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023.",

year = "2024",

month = apr,

doi = "10.1007/s00208-023-02575-0",

language = "English (US)",

volume = "388",

pages = "4037--4052",

journal = "Mathematische Annalen",

issn = "0025-5831",

publisher = "Springer New York",

number = "4",

}

TY - JOUR

T1 - A positive proportion of quartic fields are not monogenic yet have no local obstruction to being so

AU - Alpöge, Levent

AU - Bhargava, Manjul

AU - Shnidman, Ari

N1 - Publisher Copyright: © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023.

PY - 2024/4

Y1 - 2024/4

N2 - 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.

AB - 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.

UR - https://www.scopus.com/pages/publications/85156097930

UR - https://www.scopus.com/inward/citedby.url?scp=85156097930&partnerID=8YFLogxK

U2 - 10.1007/s00208-023-02575-0

DO - 10.1007/s00208-023-02575-0

M3 - Article

AN - SCOPUS:85156097930

SN - 0025-5831

VL - 388

SP - 4037

EP - 4052

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 4

ER -

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

Abstract

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this