Stickelberger ideals and divisor class numbers
type: Item
instance of: scholarly article
title: Stickelberger ideals and divisor class numbers
zbMATH Open document ID: 1034.11062
author: Linsheng Yin
published in: Mathematische Zeitschrift
publication date: 2002-06-27T00:00:00Z
review text: \textit{K. Iwasawa} introduced in [Ann. Math. (2) { 76}, 171--179 (1962; Zbl 0125.02003)], using the Stickelberger elements, the notion of Stickelberger ideal whose elements annihilate the ideal class group of the cyclotomic field of prime power conductor. Iwasawa showed that the index of the Stickelberger ideal in the minus part of the group ring is equal to the relative class number of the cyclotomic field. In [Ann. Math. (2) {108}, 107--134 (1978; Zbl 0395.12014)] \textit{W. Sinnott} extends the results of Iwasawa to a cyclotomic field by introducing cohomology to the computation of the indices. In the function field case, by Weil's theorem, partial zeta functions are rational functions of \(q ^ {-s}\) where \(q\) denotes the cardinality of the field of constants. This makes it possible to construct a larger Stickelberger ideal compared to the number field case. In the paper under review the author gives a new definition of the Stickelberger ideal. For a finite abelian extension \(K/k\) of global function fields with Galois group \(G\), using the Stickelberger elements associated to \(K/k\), studied by Tate, Deligne and Hayes, an ideal is defined, called the Stickelberger ideal of \(K\), in the integral group ring \({\mathbb Z}[G]\). The elements of the ideal annihilate the group of divisor classes of degree zero of \(K\) and the rank of the ideal is equal to the degree of the extension \(K/k\). When \(K/k\) is a (wide or narrow) ray class extension of function fields the index of the Stickelberger ideal in the integral group ring is equal to the divisor class number of \(K\) up to an explicit factor.
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R60
zbMATH DE Number: 1761104
zbMATH Keywords: Stickelberger elements
zbMATH Keywords: Stickelberger ideal
zbMATH Keywords: ray class groups
zbMATH Keywords: cyclotomic function fields
reviewed by: Gabriel D. Villa Salvador
MaRDI profile type: Publication
full work available at URL: s002090100310
OpenAlex ID: W2023002475
DOI: 10.22028/D291-26181
DOI: 10.1007/S002090100310
Recommended article: Stickelberger ideals and relative class numbers in function fields
Recommended article: A note on the index of irregularity
Recommended article: Q5441599
Recommended article: Q4454812
Recommended article: On ray class annihilators of cyclotomic function fields
Recommended article: Cyclotomic units and Stickelberger ideals of global function fields
Recommended article: An extended Stickelberger ideal of the compositum of a bicyclic field and an imaginary quadratic field
Recommended article: Q5421238
Recommended article: The indices of Stickelberger ideals of function fields
Recommended article: Q4454813
description: scientific article; zbMATH DE number 1761104