Stickelberger subideals for a prime modulus related to Kummer-type congruences
type: Item
instance of: scholarly article
title: Stickelberger subideals for a prime modulus related to Kummer-type congruences
zbMATH Open document ID: 0898.11042
author: Takashi Agoh
published in: Journal of Number Theory
publication date: 1998-10-26T00:00:00Z
review text: Let \(p\) be a prime \(\geq 5\) and let \(G\) denote a cyclic group of order \(p-1\). For a set \(\Gamma\) of different integers in the interval \([2,p-1]\), the author introduces a subideal \(\mathcal I_\Gamma (p)\) of the Stickelberger ideal \(\mathcal I(p)\) in the group ring \((\mathbb{Z}/ p\mathbb{Z})[G]\). He finds various bases (over \(\mathbb{Z}/p\mathbb{Z}\)) of this subideal. They are closely related to the following ``Kummer-type'' system of congruences: \[ \phi _{p-1}(t) \equiv 0,\quad B_{2m}^{(\Gamma)} \phi _{p-2m}(t) \equiv 0 \pmod p, \qquad (m=1,\dots,(p-3)/2). \] Here \(\phi _k(x)\) is the Mirimanoff polynomial and \(B_k^{(\Gamma)} = \prod _{N \in \Gamma} (N^k-1)B_k/k\), where \(B_k\) denotes the \(k\)-th Bernoulli number. Through an index formula the author establishes a relationship between his subideal and the relative class number of the \(p\)-th cyclotomic field. Moreover he shows that dim\(\mathcal I_\Gamma (p)\) equals \((p-1)/2\) minus the number of \(p\)-divisible \(B_{2m}^{(\Gamma)}\). According to the author, many of the ideas are adaptations or generalizations of those by \textit{L. Skula} [see, e.g., J. Number Theory 58, No. 1, 173-195 (1996; Zbl 0861.11063)]. The original Kummer system of congruences appeared in the study of Fermat's equation.
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 1109647
zbMATH Keywords: cyclotomic fields
zbMATH Keywords: Stickelberger ideals
zbMATH Keywords: Bernoulli numbers
zbMATH Keywords: system of congruences
zbMATH Keywords: Mirimanoff polynomials
zbMATH Keywords: relative class number
MaRDI profile type: Publication
full work available at URL: jnth.1997.2183
OpenAlex ID: W2068640612
cites work: Stickelberger subideals related to Kummer type congruences
cites work: Q4220431
cites work: Kummer type congruences and Stickelberger subideals
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cites work: On a special ideal contained in the Stickelberger ideal
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DOI: 10.1006/JNTH.1997.2183
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description: scientific article; zbMATH DE number 1109647