Class numbers of pure quintic fields
type: Item
instance of: scholarly article
title: Class numbers of pure quintic fields
zbMATH Open document ID: 1338.11100
author: Hirotomo Kobayashi
published in: Journal of Number Theory
publication date: 2015-12-07T00:00:00Z
review text: Suppose that \(K=\mathbb{Q}(m^{1/5})\) where \(m\) is a fifth power free integer. In the paper under review, the author proves the following nice result: If a prime \(p\equiv 4 \mod 5\) divides \(m\), then \(5\) divides the class number of \(K\). This is a particular case of a general conjecture of the author for the class number of the algebraic number field \(\mathbb Q(m^{1/\ell})\). To achieve his goal for the number field \(K\), the author obtains new results on the arithmetic (class number, unit group) of the maximal real subfield \(L^+\) of the Galois closure \(L\) of \(K\). The introduction in the paper starts with an interesting survey of previous results of \textit{T. Honda} [J. Number Theory 3, 7--12 (1971; Zbl 0222.12004)], \textit{C. J. Parry} and \textit{C. D. Walter} [Mathematika 24, 133 (1977; Zbl 0364.12006); ibid. 23, 220--226 (1976; Zbl 0364.12005)], and \textit{K. Iimura} [J. Reine Angew. Math. 292, 201--210 (1977; Zbl 0351.12003)] on the divisibility of the class number of an algebraic number field by a prime number.
reviewed by: Claude Levesque
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 11R21
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 6516879
zbMATH Keywords: pure quintic fields
zbMATH Keywords: class numbers
zbMATH Keywords: unit groups
Wikidata QID: Q114157449
MaRDI profile type: Publication
full work available at URL: j.jnt.2015.09.017
OpenAlex ID: W1938152234
cites work: Pure cubic fields whose class numbers are multiples of three
cites work: A criterion for the class number of a pure quintic field to be divisible by 5.
cites work: Class numbers of algebraic number fields of Eisenstein type
cites work: Über die Klassenzahlen algebraischer Zahlkörper
cites work: Q4097368
cites work: The class number of pure fields of prime degree
cites work: Q5690489
DOI: 10.1016/J.JNT.2015.09.017
description: scientific article; zbMATH DE number 6516879