Dihedral side extensions and class groups
type: Item
instance of: scholarly article
title: Dihedral side extensions and class groups
zbMATH Open document ID: 1213.11188
author: David Hubbard
published in: Journal of Number Theory
publication date: 2008-12-05T00:00:00Z
review text: Let \(p\) be an odd prime, let \(L/\mathbb Q\) be an extension of degree \(p\) whose normal closure \(K/\mathbb Q\) has Galois group \(D_p\), the dihedral group of order \(2p\), and let \(F\) denote the unique quadratic subfield of \(K\). The author proves that if there is exactly one ramified prime in \(K/F\), then the ranks of the \(p\)-class groups of \(F\) and \(L\) are related by the inequalities \[ \text{rk\;Cl}_p(F) \leq \text{rk\;Cl}_p(L) \leq \frac{p-1}2 \text{rk\;Cl}_p(F). \] This generalizes a result due to \textit{T. Callahan} [Mathematika 21(1974), 168--188 (1975; Zbl 0312.12009)] according to which the ranks of the \(3\)-class groups of \(F\) and \(L\) coincide. Results similar to those proved here were derived by \textit{R. Bölling} [Math. Nachr. 135, 275--310 (1988; Zbl 0674.12003)] in the case of unramified extensions, which in turn were generalized by the reviewer [Math. Nachr. 278, No. 6, 679--691 (2005; Zbl 1067.11069)] to general base fields with class number not divisible by \(p\).
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 5378822
zbMATH Keywords: dihedral extensions
zbMATH Keywords: class groups
zbMATH Keywords: reflection theorems
zbMATH Keywords: class field theory
zbMATH Keywords: ambiguous classes
describes a project that uses: KANT/KASH
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: j.jnt.2007.04.008
OpenAlex ID: W2013922534
cites work: The Magma algebra system. I: The user language
cites work: KANT V4
cites work: Ranks of 3-class groups of non-Galois cubic fields
cites work: Sur les \(l\)-classes d'ideaux des extensions non galoisiennes de Q de degré premier impair \(l\) à cloture galoisienne diedrale de degré 2\(l\)
cites work: A note on class numbers of algebraic number fields
DOI: 10.1016/J.JNT.2007.04.008
description: scientific article; zbMATH DE number 5378822