Demjanenko matrix and recursion formula for relative class number over function fields.
type: Item
instance of: scholarly article
title: Demjanenko matrix and recursion formula for relative class number over function fields.
zbMATH Open document ID: 1057.11057
DOI: 10.1016/S0022-314X(02)00023-9
author: Hwanyup Jung
author: Jaehyun Ahn
published in: Journal of Number Theory
publication date: 2003-04-28T00:00:00Z
review text: Let \(M=P^n\), where \(P\in\mathbb F_q[T]\) is irreducible, let \(K_M\) be the \(M\)th cyclotomic function field and \(K_M^+\) its maximal real subfield. Let \(h^-(\mathcal O_{K_M})=h(\mathcal O_{K_M})/h(\mathcal O_{K_M^+})\) denote its relative ideal class number. \textit{S. Bae} and \textit{P. Kang} [Acta Arith. 102, No. 3, 251--259 (2002; Zbl 0989.11064)] have defined analogues of the classical Maillet and Demyanenko matrices. The present authors give a new analogue of the latter, and show that its determinant equals \(h^-(\mathcal O_{K_M})\). Further, they give a recursion formula which enables one to compute the value of this determinant. There are results about the highest power of a prime divisor of \(q-1\) dividing \(h^-(\mathcal O_{K_M})\). There are also several numerical examples.
Mathematics Subject Classification ID: 11R60
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 1902867
zbMATH Keywords: cyclotomic function field
zbMATH Keywords: Demyanenko matrix
zbMATH Keywords: relative class number
zbMATH Keywords: recursion formula
reviewed by: Veikko Ennola
MaRDI profile type: Publication
cites work: Cyclotomic units and Stickelberger ideals of global function fields
cites work: Class numbers of cyclotomic function fields
cites work: A recursion formula for the relative class number of the \(p^ n\)-th cyclotomic field
cites work: Demjanenko matrix, class number, and Hodge group
cites work: Class Numbers of Cyclotomic Function Fields
cites work: On the Stickelberger ideal and the circular units of an abelian field
cites work: The Stark conjectures on Artin \(L\)-functions at \(s=0\). Lecture notes of a course in Orsay edited by Dominique Bernardi and Norbert Schappacher.
cites work: Stickelberger ideals and relative class numbers in function fields
full work available at URL: s0022-314x(02)00023-9
OpenAlex ID: W1969304369
description: scientific article; zbMATH DE number 1902867