Power bases for cyclotomic integer rings
type: Item
instance of: scholarly article
title: Power bases for cyclotomic integer rings
zbMATH Open document ID: 0923.11150
author: L. Robertson
published in: Journal of Number Theory
publication date: 1999-10-13T00:00:00Z
review text: Let \(p\) be an odd prime, \(\xi\) a primitive \(p\)-th root of unity and \({\mathcal O}_p\) the ring of integers of the cyclotomic field \(\mathbb{Q}(\xi)\). It is a classical problem to investigate power integral bases in \(\mathbb{Q}(\xi)\), that is to determine all \(\alpha\) such that \({\mathcal O}_p=\mathbb{Z}[\alpha]\) or equivalently \(\{ 1,\alpha,\ldots,\alpha^{p-1}\}\) is an integral basis. \textit{A. Bremner} [J. Number Theory 28, 288-298 (1988; Zbl 0637.12001)] conjectured that, up to translations and conjugation only \(\alpha=\xi,1/(1+\xi)\) generate power integral bases. Bremner proved this conjecture for \(p=7\). The author establishes a criterion for Bremner's conjecture for a given regular prime \(p\). The proof of this criterion is based on a determinant formula for the relative class number of \(\mathbb{Q}(\xi)\). This criterion is then used to verify the conjecture for \(p\leq 23\), \(p\neq 17\).
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R04
zbMATH DE Number: 1207955
zbMATH Keywords: power integral bases
zbMATH Keywords: cyclotomic fields
zbMATH Keywords: Bremner's conjecture
reviewed by: István Gaál
MaRDI profile type: Publication
full work available at URL: jnth.1997.2214
OpenAlex ID: W1971717009
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DOI: 10.1006/JNTH.1997.2214
description: scientific article; zbMATH DE number 1207955