Divisor class number one problem for Abelian extensions over rational function fields
type: Item
instance of: scholarly article
title: Divisor class number one problem for Abelian extensions over rational function fields
zbMATH Open document ID: 1183.11072
author: Hwanyup Jung
author: Jaehyun Ahn
published in: Journal of Algebra
publication date: 2007-04-18T00:00:00Z
review text: The characterization of cyclotomic function fields and their maximal real subfields with divisor class number one was done by \textit{M. Kida} and \textit{N. Murabayashi} [Tokyo J. Math. 14, 45--56 (1991; Zbl 0734.11060)]. Also \textit{S. Bae} and \textit{P. Kang} [Acta Arith. 102, 251--259 (2002; Zbl 0989.11064)] determined all cyclotomic function fields with relative divisor class number one. This paper determines all abelian extensions of rational function fields with divisor class number one and all imaginary abelian extensions with relative divisor class number one. Let \(k=\mathbb{F}_q(t)\) be the rational function field over the finite field \(\mathbb{F}_q\) with \(q\geq 3\) elements. The author relies on Proposition \(1\) and \(2\) in [Tokyo J. Math. 14, 45--56 (1991; Zbl 0734.11060)] to reduce the problem to finding abelian extensions over \(k\) with genus \(0\) or \(1\). The computation of the genus of a finite abelian extension \(K\) of \(k\) makes use of the Riemann-Hurwitz formula and the relation between the different of \(K/k\) and the ramification groups of \(K/k\). First the author computes all imaginary abelian extensions \(K/k\) with divisor class number one in the cases the genus \(g_K\) of \(K\) is zero and one respectively. Then the author computes all real abelian extensions \(K/k\) with genus zero.
reviewed by: Jing Long Hoelscher
Mathematics Subject Classification ID: 11R60
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 5144376
zbMATH Keywords: cyclotomic function field
zbMATH Keywords: divisor class number
zbMATH Keywords: genus
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.02.006
OpenAlex ID: W1995521094
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DOI: 10.1016/J.JALGEBRA.2003.02.006
description: scientific article; zbMATH DE number 5144376