Class numbers and units of E. Lehmer's cyclic quintic fields
type: Item
instance of: scholarly article
title: Class numbers and units of E. Lehmer's cyclic quintic fields
zbMATH Open document ID: 0865.11070
author: Stéphane Jeannin
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1996-11-24T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/233472
full work available at URL: jtnb8-1.html
review text: In her article `Connection between Gaussian periods and cyclic units' [Math. Comput. 50, 535-541 (1988; Zbl 0652.12004)], \textit{E. Lehmer} introduced a parametrized family of quintic polynomials \(P_t\) whose roots generate a subgroup of finite index in the unit group of the corresponding cyclic quintic field. In the article under review, the author proves that these units are fundamental if the discriminant of the polynomial satisfies a certain condition (this generalizes previous results of \textit{R. Schoof} and \textit{L. C. Washington} [Math. Comput. 50, 543-556 (1988; Zbl 0649.12007)]), and computes the ideal class groups of the fields for small values of \(t\) and conductor \(<3 \cdot 10^6\). This is done by first computing the group of cyclotomic units, and then checking which of the units in this group are powers of units in the field.
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11R21
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11Y40
zbMATH DE Number: 912499
zbMATH Keywords: simplest fields
zbMATH Keywords: class number
zbMATH Keywords: fundamental units
zbMATH Keywords: cyclic quintic field
zbMATH Keywords: ideal class groups
zbMATH Keywords: cyclotomic units
MaRDI profile type: Publication
OpenAlex ID: W2043661610
cites work: Q4132630
cites work: Q5454198
cites work: Cyclotomy and Delta Units
cites work: Connection Between Gaussian Periods and Cyclic Units
cites work: Quintic Polynomials and Real Cyclotomic Fields with Large Class Number
DOI: 10.5802/JTNB.157
description: scientific article; zbMATH DE number 912499