Note on the number of integral ideals in Galois extensions
type: Item
instance of: scholarly article
title: Note on the number of integral ideals in Galois extensions
zbMATH Open document ID: 1273.11160
author: Guangshi Lü
author: Yonghui Wang
published in: Science China. Mathematics
publication date: 2011-02-25T00:00:00Z
review text: Let \(K\) be a number field and \(a_k\) be the number of integral ideals in \(K\) with norm \(k\). Landau studied the sum \(\sum_{k \leq x} a_k\) when \(K\) has degree \(\geq 2\). Several other authors have studied this problem. The best known results for number fields \(K\) of degree \(\geq 3\) is by \textit{W. G. Nowak} [Math. Nachr. 161, 59--74 (1993; Zbl 0803.11061)]. In this paper, the authors study the sum \(\sum_{k \leq x} a_k^l\) for any integer \(l \geq 2\) and when \(K\) is Galois of degree \(n \geq 2\). Their result improves an earlier result of \textit{K. Chandrasekharan} and \textit{A. Good} [Monatsh. Math. 95, 99--109 (1983; Zbl 0498.12009)]. Furthermore, if \(K\) is abelian, the authors obtain a better bound. The authors also study the number of solutions of polynomial congruences as an application of their results.
reviewed by: Sanoli Gun
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11R45
zbMATH DE Number: 5857698
zbMATH Keywords: Dedekind zeta-function
zbMATH Keywords: power sum of integral ideals
zbMATH Keywords: polynomial congruence
MaRDI profile type: Publication
full work available at URL: s11425-010-4091-7
OpenAlex ID: W2131864709
cites work: On the number of integral ideals in Galois extensions
cites work: The approximate functional equation for a class of zeta-functions
cites work: On the Sum ∑k=1xd(f(k))
cites work: THE TWELFTH POWER MOMENT OF THE RIEMANN-FUNCTION
cites work: The growth rate of the Dedekind Zeta-function on the critical line
cites work: The number of ideals in a quadratic field. II
cites work: A mean value theorem for Dirichlet series and a general divisor problem
cites work: Functoriality and number of solutions of congruences
cites work: The mean number of solutions of certain congruences
cites work: Q3220660
cites work: Q4830109
cites work: Q3324043
cites work: On the distribution of ideals in cubic number fields
cites work: On the Distribution of Integer Ideals in Algebraic Number Fields
cites work: Q3982414
cites work: Higher moments of certain \(L\)-functions on the critical line
cites work: Q4528915
DOI: 10.1007/S11425-010-4091-7
Recommended article: On the number of integral ideals in a number field
Recommended article: A Note on Galois Cohomology of Algebraic Integers
Recommended article: Counting integral ideals in a number field
Recommended article: Q4863887
Recommended article: Q4767377
Recommended article: Some facts concerning integral representations of the ideals in an algebraic number field
Recommended article: On the Distribution of Integer Ideals in Algebraic Number Fields
Recommended article: Note on the rings of integers of certain tame 2-Galois extensions over a number field
Recommended article: On the number of integral ideals in two different quadratic number fields
Recommended article: On counting rings of integers as Galois modules
description: scientific article; zbMATH DE number 5857698