A note on the index of irregularity
type: Item
instance of: scholarly article
title: A note on the index of irregularity
zbMATH Open document ID: 0589.12006
DOI: 10.1016/0022-314X(86)90064-8
published in: Journal of Number Theory
publication date: 1986-01-01T00:00:00Z
review text: Let \(\ell\) be an odd prime and let G be the Galois group over \({\mathbb{Q}}\) of the \(\ell\)-th cyclotomic field. Let \(R^-(\ell)\) be the subring of the group ring of G over \({\mathbb{Z}}/\ell {\mathbb{Z}}\) which is the -1- eigenspace for complex conjugation. There is an ideal \(I^-(\ell)\) of \(R^-(\ell)\), called the Stickelberger ideal. In the case where the group ring is taken over \({\mathbb{Z}}\), \textit{K. Iwasawa} [Ann. Math., II. Ser. 76, 171-179 (1962; Zbl 0125.020)] showed that the index \([R^- :I^-]\) of the Stickelberger ideal is \(h^-\), the relative class number of the \(\ell\)-th cyclotomic field. In the present case, the author shows that the index is \(\ell^ i\), where i is the index of irregularity of \(\ell\). He also studies various properties of the Stickelberger ideal mod \(\ell\).
reviewed by: Lawrence C. Washington
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 3946270
zbMATH Keywords: cyclotomic field
zbMATH Keywords: group ring
zbMATH Keywords: Stickelberger ideal
zbMATH Keywords: index
zbMATH Keywords: relative class number
zbMATH Keywords: index of irregularity
author: Ladislav Skula
MaRDI profile type: Publication
cites work: Maillet’s determinant
cites work: Q5677573
cites work: A class number formula for cyclotomic fields
cites work: Q3874311
cites work: Index of irregularity of a prime.
cites work: Another proof of Iwasawa's class number formula
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description: scientific article; zbMATH DE number 3946270