The density of discriminants of quintic rings and fields
type: Item
instance of: scholarly article
title: The density of discriminants of quintic rings and fields
zbMATH Open document ID: 1220.11139
author: Manjul Bhargava
published in: Annals of Mathematics. Second Series
publication date: 2011-02-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1005.5578
review text: Let \(N_5^{(i)}(\xi,\eta)\) denote the number of quintic algebraic number fields \(K\), up to isomorphism, having \(5-2i\) real embeddings and satisfying \(\xi<\text{Disc}(K)<\eta\). The author proves that \[ \lim_{X\to\infty}\frac{N_5^{(0)}(0,X)}{X}=\frac{1}{240}\prod_p (1+p^{-2}-p^{-4}-p^{-5}); \] \[ \lim_{X\to\infty}\frac{N_5^{(1)}(-X,0)}{X}=\frac{1}{24}\prod_p (1+p^{-2}-p^{-4}-p^{-5}); \] \[ \lim_{X\to\infty}\frac{N_5^{(2)}(0,X)}{X}=\frac{1}{16}\prod_p (1+p^{-2}-p^{-4}-p^{-5}). \] Let \(M_5^{(i)}(\xi,\eta)\) denote the number of isomorphism classes of orders \({\mathcal O}\) in quintic fields having \(5-2i\) real embeddings and satisfying \(\xi<\text{Disc}\,{\mathcal O}<\eta\). Then there exists a positive constant \(\alpha\) such that \[ \lim_{X\to\infty}\frac{M_5^{(0)}(0,X)}{X}=\frac{\alpha}{240}; \] \[ \lim_{X\to\infty}\frac{M_5^{(1)}(-X,0)}{X}=\frac{\alpha}{24}; \] \[ \lim_{X\to\infty}\frac{M_5^{(2)}(0,X)}{X}=\frac{\alpha}{16}. \]
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11R45
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 5850182
zbMATH Keywords: density theorems
zbMATH Keywords: discriminants
zbMATH Keywords: number fields
zbMATH Keywords: quintic fields
MaRDI profile type: Publication
OpenAlex ID: W2950749849
Wikidata QID: Q30053312
arXiv ID: 1005.5578
cites work: Higher composition laws. IV: The parametrization of quintic rings
cites work: The density of discriminants of quartic rings and fields
cites work: Mass Formulae for Extensions of Local Fields, and Conjectures on the Density of Number Field Discriminants
cites work: Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves
cites work: On the Davenport-Heilbronn theorems and second order terms
cites work: On a Principle of Lipschitz
cites work: On the Class-Number of Binary Cubic Forms (I)
cites work: On the density of discriminants of cubic fields. II
cites work: The adelic zeta function associated to the space of binary cubic forms. II: Local theory.
cites work: Q5622438
cites work: A classification of irreducible prehomogeneous vector spaces and their relative invariants
cites work: On zeta functions associated with prehomogeneous vector spaces
cites work: On Dirichlet series whose coefficients are class numbers of integral binary cubic forms
cites work: Prehomogeneous vector spaces and field extensions
cites work: Q4282628
DOI: 10.4007/ANNALS.2010.172.1559
description: scientific article