A Demjanenko matrix for abelian fields of prime power conductor
type: Item
instance of: scholarly article
title: A Demjanenko matrix for abelian fields of prime power conductor
zbMATH Open document ID: 0829.11054
author: Jonathan W. Sands
author: Wolfgang Schwarz
published in: Journal of Number Theory
publication date: 1996-01-22T00:00:00Z
review text: Fix a power \(q = p^r\) of an odd prime \(p\), and write \(G = (\mathbb{Z}/q \mathbb{Z})^*\), \(M = \{\overline a \in G : 0 < a \leq (q - 1)/2\}\). Let \(c_M\) be the characteristic function of \(M\). The corresponding Demyanenko matrix is \(D_q = (c_M (\overline a \overline b))_{\overline a, \overline b \in M}\). The determinant of \(D_q\) provides a formula for the relative class number of \(\mathbb{Q} (\zeta_q)\). The goal of the paper is to extend the theory to allow consideration of a quotient group of \(G\) and the corresponding subfield \(K\) of \(\mathbb{Q} (\zeta_q)\). The connection between the Demyanenko matrix and the real cyclotomic units leads to a simple proof of a parity result due to Hasse and Garbanati. The authors establish an expression for the relative class number \(h^- (K)\) as the determinant of a matrix with relatively small integer entries. This leads to a reasonable upper bound for \(h^- (K)\).
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 765163
zbMATH Keywords: Abelian field of prime power conductor
zbMATH Keywords: Demyanenko matrix
zbMATH Keywords: relative class number
zbMATH Keywords: upper bound
reviewed by: Veikko Ennola
MaRDI profile type: Publication
full work available at URL: jnth.1995.1057
OpenAlex ID: W2056666006
DOI: 10.1006/JNTH.1995.1057
Recommended article: On Demjanenko's matrix and Maillet's determinant for imaginary abelian number fields
Recommended article: Q4353513
Recommended article: Demjanenko matrix for imaginary abelian fields of odd conductors
Recommended article: The Conductor Density of Abelian Number Fields
Recommended article: On the class number of an abelian field with prime conductor
Recommended article: Structure and prime decomposition law and relative extensions of abelian fields with prime power degree
Recommended article: Q3144692
Recommended article: Matrix powers over finite fields
Recommended article: On a conjecture about the generalized exponent of primitive matrices
Recommended article: A note on integral matrices associated to finite fields
description: scientific article; zbMATH DE number 765163