On the class numbers of certain Hilbert class fields
type: Item
instance of: scholarly article
title: On the class numbers of certain Hilbert class fields
zbMATH Open document ID: 0806.11053
DOI: 10.1007/BF02568352
author: Akito Nomura
published in: Manuscripta Mathematica
publication date: 1995-02-13T00:00:00Z
full work available at URL: https://eudml.org/doc/155848
review text: The main theorem, which is a refinement of the author's earlier work [Osaka J. Math. 28, 55-62 (1991; Zbl 0722.11055)] states: Let \(p\) be an odd prime and assume the Galois extensions of number fields \(L/K/k\) satisfy: (1) the degree \([K:k]\) is prime to \(p\); (2) \(L/K\) is an unramified \(p\)-extension. Let \(\varepsilon: 1\to \mathbb{Z}/ p\mathbb{Z}\to E\to \text{Gal} (L/k)\to 1\) be a non-split central embedding extension. Then there exists a Galois extension \(M/k\) such that (i) \(M/k\) gives a proper solution to the central embedding problem \((L/k, \varepsilon)\) and (ii) \(M/k\) is unramified. As an application of the main theorem, the author obtains the following result: Let \(l\) and \(p\) be distinct odd primes and assume that \(p\) has even order modulo \(l\). Let \(K/k\) be an abelian \(l\)-extension. If the class number of \(K\) is divisible by \(p\), then the class number of the maximal unramified abelian \(p\)-extension over \(K\) is also divisible by \(p\).
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 492754
zbMATH Keywords: Hilbert class fields
zbMATH Keywords: Galois extension
zbMATH Keywords: central embedding problem
zbMATH Keywords: class number
zbMATH Keywords: abelian \(p\)-extension
reviewed by: Charles J. Parry
MaRDI profile type: Publication
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OpenAlex ID: W1994365115
description: scientific article; zbMATH DE number 492754