An improvement on Schmidt's bound on the number of number fields of bounded discriminant and small degree

Manjul Bhargava; Arul Shankar; Xiaoheng Wang

doi:10.1017/fms.2022.71

An improvement on Schmidt's bound on the number of number fields of bounded discriminant and small degree

Manjul Bhargava
, Arul Shankar
, Xiaoheng Wang

Mathematics

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

5 Scopus citations

Abstract

We prove an improvement on Schmidt's upper bound on the number of number fields of degree n and absolute discriminant less than X for 6 ≤ n ≤ 94. We carry this out by improving and applying a uniform bound on the number of monic integer polynomials, having bounded height and discriminant divisible by a large square, that we proved in a previous work [7].

Original language	English (US)
Article number	71
Journal	Forum of Mathematics, Sigma
Volume	10
DOIs	https://doi.org/10.1017/fms.2022.71
State	Published - Oct 10 2022

All Science Journal Classification (ASJC) codes

Analysis
Theoretical Computer Science
Algebra and Number Theory
Statistics and Probability
Mathematical Physics
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Mathematics

Access to Document

10.1017/fms.2022.71

Cite this

@article{a3db2192a1854e3b93e2765eb2c3e44e,

title = "An improvement on Schmidt's bound on the number of number fields of bounded discriminant and small degree",

abstract = "We prove an improvement on Schmidt's upper bound on the number of number fields of degree n and absolute discriminant less than X for 6 ≤ n ≤ 94. We carry this out by improving and applying a uniform bound on the number of monic integer polynomials, having bounded height and discriminant divisible by a large square, that we proved in a previous work [7].",

author = "Manjul Bhargava and Arul Shankar and Xiaoheng Wang",

note = "Publisher Copyright: {\textcopyright} ",

year = "2022",

month = oct,

day = "10",

doi = "10.1017/fms.2022.71",

language = "English (US)",

volume = "10",

journal = "Forum of Mathematics, Sigma",

issn = "2050-5094",

publisher = "Cambridge University Press",

}

TY - JOUR

T1 - An improvement on Schmidt's bound on the number of number fields of bounded discriminant and small degree

AU - Bhargava, Manjul

AU - Shankar, Arul

AU - Wang, Xiaoheng

PY - 2022/10/10

Y1 - 2022/10/10

N2 - We prove an improvement on Schmidt's upper bound on the number of number fields of degree n and absolute discriminant less than X for 6 ≤ n ≤ 94. We carry this out by improving and applying a uniform bound on the number of monic integer polynomials, having bounded height and discriminant divisible by a large square, that we proved in a previous work [7].

AB - We prove an improvement on Schmidt's upper bound on the number of number fields of degree n and absolute discriminant less than X for 6 ≤ n ≤ 94. We carry this out by improving and applying a uniform bound on the number of monic integer polynomials, having bounded height and discriminant divisible by a large square, that we proved in a previous work [7].

UR - https://www.scopus.com/pages/publications/85140242348

UR - https://www.scopus.com/pages/publications/85140242348#tab=citedBy

U2 - 10.1017/fms.2022.71

DO - 10.1017/fms.2022.71

M3 - Article

AN - SCOPUS:85140242348

SN - 2050-5094

VL - 10

JO - Forum of Mathematics, Sigma

JF - Forum of Mathematics, Sigma

M1 - 71

ER -

An improvement on Schmidt's bound on the number of number fields of bounded discriminant and small degree

Abstract

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this