Remarks on unit indices of imaginary abelian number fields
type: Item
instance of: scholarly article
title: Remarks on unit indices of imaginary abelian number fields
zbMATH Open document ID: 0654.12002
DOI: 10.1007/BF01258662
author: Mikihito Hirabayashi
author: Ken-ichi Yoshino
published in: Manuscripta Mathematica
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155291
review text: Let \(K\) be an imaginary abelian field. Within its unit group, the real units together with the roots of unity generate a subgroup of index \(Q_K=1\) or 2. This number is called the unit index of \(K\). A result proved by \textit{H. Hasse} in his monograph ``Über die Klassenzahl abelscher Zahlkörper'' [Berlin: Akademie-Verlag (1952; Zbl 0046.26003); reprint (1985; Zbl 0668.12003)] states that \(Q_K=1\) if \(f\), the conductor of \(K\), is a prime power. The present authors determine \(Q_ K\) in cases \(f\) is \(4p^a\), \(p^aq^b\) or \(2^np^a\) \((n\geq 3)\), where \(p\) and \(q\) are different odd primes and, in the third case, \(8\nmid p-1\). They also study \(Q_K\) for \(f=8p\) with \(8| p-1\), and for \(f=4pq\). The results provide many examples of cases in which \(Q_K=1\) but \(K\) contains a subfield \(k\) with \(Q_k=2\). It is pointed out how one should modify the places in Hasse's book [op. cit.] where it is erroneously assumed that \(Q_K\) be divisible by \(Q_k\).
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 4067039
zbMATH Keywords: class numbers
zbMATH Keywords: real units
zbMATH Keywords: unit index
zbMATH Keywords: counterexamples to Satz 29 in Hasse's monograph
MaRDI profile type: Publication
cites work: Q5512252
cites work: Q5843509
cites work: The unit indices of imaginary abelian number fields
cites work: A note on class numbers of algebraic number fields
cites work: Q3943047
description: scientific article; zbMATH DE number 4067039