On finiteness of the numbers of Euclidean fields in some classes of number fields
type: Item
instance of: scholarly article
title: On finiteness of the numbers of Euclidean fields in some classes of number fields
zbMATH Open document ID: 0548.12004
author: Shigeki Egami
published in: Tokyo Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: An ideal class C of a number field K is called norm-Euclidean if there is an ideal \({\mathfrak a}\) such that for every \(\alpha \in K\) there exists a \(\beta \in {\mathfrak a}\) such that \(| N_ K(\alpha -\beta)| <N_ K {\mathfrak a}.\) A field is Euclidean if and only if there is one ideal class and that is norm-Euclidean. Let \(F_ 1^{(n)}\) be the set of cyclic fields of degree n such that every rational prime factor of the discriminant is totally ramified, let \(F_ 2^{(\ell)}\) be \(\{{\mathbb{Q}}(^{\ell}\sqrt{m};\quad m\in {\mathbb{Z}}\},\) and let \(F_ 3^{(d)}\) be the set of non-Galois cubic fields K such that the Galois closure of K/\({\mathbb{Q}}\) contains \({\mathbb{Q}}(\sqrt{d}).\) By means of estimates of character sums it is shown that \(F_ 1^{(n)}\) and \(F_ 3^{(d)}\) contain only finitely many fields with a norm- Euclidean ideal class: the same is true for \(F_ 2^{(\ell)}\) under the assumption of the Generalized Riemann Hypothesis. Moreover \(F_ 2^{(5)}\) contains only finitely many Euclidean fields, without assuming the G.R.H..
Mathematics Subject Classification ID: 11R21
Mathematics Subject Classification ID: 11L40
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 3873448
zbMATH Keywords: norm-Euclidean ideal class
zbMATH Keywords: cyclic fields
zbMATH Keywords: discriminant
zbMATH Keywords: non-Galois cubic fields
zbMATH Keywords: estimates of character sums
zbMATH Keywords: Generalized Riemann Hypothesis
zbMATH Keywords: finitely many Euclidean fields
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270153001
OpenAlex ID: W2019365412
DOI: 10.3836/TJM/1270153001
description: scientific article; zbMATH DE number 3873448