On divisibility of class number of real abelian fields of prime conductor
type: Item
instance of: scholarly article
title: On divisibility of class number of real abelian fields of prime conductor
zbMATH Open document ID: 0788.11052
DOI: 10.1007/BF02941333
author: Stanislav Jakubec
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1994-02-06T00:00:00Z
review text: For an odd prime \(p\), let \(\zeta_ p= e^{2\pi i/p}\) and denote by \(K\) the subfield of the real cyclotomic field \(\mathbb{Q}(\zeta_ p+ \zeta_ p^{-1})\) whose degree over \(\mathbb{Q}\) is a prime, say \(\ell\). This paper is a study of the divisibility of \(h_ K\), the class number of \(K\), by odd primes \(q<p\). Let \(C(K)\) denote the group of cyclotomic units of \(K\). The author first uses an arbitrary generator \(\gamma\) of \(C(K)\) to produce an element \(\rho\) of \(\mathbb{Z}[\zeta_ \ell]\) whose (absolute) norm is divisible by the primes \(q\) dividing \(h_ K\). By choosing \(\gamma\) as the ``standard'' generator of \(C(K)\), he is then able to write down an explicit expression for the corresponding \(\rho\). An inspection of that expression yields a series of smooth conditions for \(h_ K\not\equiv 0\pmod q\). Three examples: \(p=2\ell+1\) and \(\text{ord}_ \ell (q)= \ell- 1\); \(p= 4\ell+1\), \(q=3\) or 5 and \(\text{ord}_ \ell(q)= \ell-1\); \(p=2\ell+1\equiv 5\pmod 9\), \(q=3\) and \(\text{ord}_ \ell(3)= (\ell- 1)/2\). (Here \(\text{ord}_ \ell(q)\) denotes the exponent \(\text{mod }\ell\) of \(q\)).
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 492584
zbMATH Keywords: abelian extensions
zbMATH Keywords: divisibility
zbMATH Keywords: class number
zbMATH Keywords: cyclotomic units
MaRDI profile type: Publication
cites work: Q3943047
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cites work: On the parity of the class number of the field of q-th roots of unity
description: scientific article; zbMATH DE number 492584