Genus theory of global fields
type: Item
instance of: scholarly article
title: Genus theory of global fields
zbMATH Open document ID: 0982.11062
author: Bruno Anglès
author: Jean-François Jaulent
published in: Manuscripta Mathematica
publication date: 2002-03-27T00:00:00Z
review text: The authors present a unified treatment of genus theory for finite extensions of global fields (number fields or function fields of curves over finite fields), which need not be Galois. For function fields \(K\), one first needs to declare a fixed place to be ``at infinity''; one can then introduce the analogue of sign functions, which generalizes the notion of leading coefficient of a polynomial. The quotient of \(K^*\) by the subgroup of its totally positive elements (w.r.t.\ the sign functions) provides the link between ideal classes and narrow ideal classes. The direct sum of this group with the group of ideals is called the group of signed ideals (= divisors); its classes form precisely the restricted class group; one can equally well define all of this for \(S\)-ideal classes, where \(S\) is a fixed set of finite places of \(K\). The \(S\)-genus field \(G\) of a finite extension \(L\) of \(K\) is the largest subfield of the \(S\)-Hilbert class field of \(L\) that is abelian over \(K\). The authors establish a generalization of a formula of \textit{Y.~Furuta} [Nagoya Math. J. 29, 281-285 (1967; Zbl 0166.05901)] for Galois extensions and \textit{L. J.~Goldstein} [Nagoya Math. J. 45, 119-127 (1972; Zbl 0234.12002)] for abelian extensions that expresses the degree of \(LH\) over \(L\) purely in terms of data associated to the extension \(L/K\) (\(S\)-class number of \(K\), number of \(S\)-units and \(S\)-local norms from \(L/K\) and data associated to the maximal abelian subextension of \(L/K\) and its localizations). Finally, the Galois group of \(G\) over the Hilbert class field of \(K\) can be determined likewise by purely local data. The formulas are made more explicit in case \(K\) is a function field with one place at infinity. Finally, the \(S\)-central class field of \(L/K\) is defined and for Galois extensions, its degree determined by establishing a link with \(H\).
reviewed by: Gunther Cornelissen
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 1463442
zbMATH Keywords: global field
zbMATH Keywords: genus theory for finite extensions of fields
zbMATH Keywords: number fields
zbMATH Keywords: function fields
zbMATH Keywords: curves over finite fields
zbMATH Keywords: analogue of sign functions
zbMATH Keywords: totally positive elements
zbMATH Keywords: narrow ideal classes
zbMATH Keywords: group of signed ideals
zbMATH Keywords: restricted class group
zbMATH Keywords: Hilbert class field
zbMATH Keywords: class number
zbMATH Keywords: local norms
MaRDI profile type: Publication
full work available at URL: s002290050229
OpenAlex ID: W1989509021
DOI: 10.1007/S002290050229
description: scientific article; zbMATH DE number 1463442