Evaluation of the Dedekind zeta functions at \(s=-1\) of the simplest quartic fields
type: Item
instance of: scholarly article
title: Evaluation of the Dedekind zeta functions at \(s=-1\) of the simplest quartic fields
zbMATH Open document ID: 1305.11093
author: D. Kharzeev
published in: Journal of Number Theory
publication date: 2014-09-02T00:00:00Z
review text: The author considers cyclic quartic fields \(K_t=\mathbb Q(\theta_t)\), \(\theta_t\) being a zero of \(X^4-tX^3-6X^2+tX+1\), where \(t\) is a positive integer such that \(t^2+16\) has no odd square factor (such fields are called simple quartic fields). \textit{C. L. Siegel} [Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl. 1969, 87--102 (1969; Zbl 0186.08804)] gave a formula for the value of the Dedekind zeta-function \(\zeta_K(s)\) of totally real fields \(K\) at negative rational integers, which for quartic fields implies \[ \zeta_K(-1)={1\over30}\sum_a\sum_{I|a\delta_K}N(I), \] where \(\delta_K\) is the different of \(K\), and \(a\) runs over totally positive elements of \(\delta_K^{-1}\) of trace \(1\). The author presents a geometric method of calculating the right hand-side of this formula, and presents a table of \(\zeta_{K_t}(-1)\) for \(t=1,2,\dots,21\).
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11R80
zbMATH DE Number: 6338993
zbMATH Keywords: quartic fields
zbMATH Keywords: Dedekind zeta function
zbMATH Keywords: totally real field
zbMATH Keywords: cyclic fields
MaRDI profile type: Publication
full work available at URL: j.jnt.2014.03.011
OpenAlex ID: W2039842252
cites work: Class number 1 criteria for real quadratic fields of Richaud-Degert type
cites work: Class number 2 criteria for real quadratic fields of Richaud-Degert type
cites work: Evaluation of the Dedekind zeta functions of some non-normal totally real cubic fields at negative odd integers
cites work: Handbook of K-Theory
cites work: \(K_i\) groups of rings of algebraic integers
cites work: Values of zeta functions and class number 1 criterion for the simplest cubic fields
cites work: Evaluation of zeta function of the simplest cubic field at negative odd integers
cites work: On the class number and unit index of simplest quartic fields
cites work: Class number one criterion for some non-normal totally real cubic fields
cites work: Class fields of abelian extensions of \(\mathbb Q\)
cites work: Classical theory of algebraic numbers
cites work: Q5579662
cites work: Q4373605
cites work: The Iwasawa conjecture for totally real fields
cites work: On the values at negative integers of the zeta-function of a real quadratic field
cites work: Q5732722
DOI: 10.1016/J.JNT.2014.03.011
description: scientific article; zbMATH DE number 6338993