Real quadratic fields with abelian 2-class field tower
type: Item
instance of: scholarly article
title: Real quadratic fields with abelian 2-class field tower
zbMATH Open document ID: 0919.11073
author: Elliot Benjamin
author: Franz Lemmermeyer
author: C. Snyder
published in: Journal of Number Theory
publication date: 1999-08-16T00:00:00Z
review text: The authors determine all real quadratic fields \(K\) for which the 2-class field tower has length at most 1, i.e., for which the 2-Hilbert class field has odd class number. By a result of \textit{R. J. Bond} [J. Number Theory 30, 1-10 (1988; Zbl 0649.12008)], the 2-class group of such \(K\) has 2-rank at most 2, and the case of interest is when it is equal to 2. In this case the number \(t\) of distinct prime factors of the discriminant of \(K\) is 3 or 4. For \(t=3\), the classification obtained is in terms of the relative quadratic and biquadratic behavior of these prime factors and the norm of the fundamental unit of \(K\). For \(t=4\), the non-trivial relation between the ideal classes of the ramified primes of \(K\) is also of importance.
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 1247918
zbMATH Keywords: quadratic field
zbMATH Keywords: 2-class field tower
zbMATH Keywords: odd class number
reviewed by: Peter Stevenhagen
MaRDI profile type: Publication
OpenAlex ID: W2083567129
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DOI: 10.1006/JNTH.1998.2291
description: scientific article; zbMATH DE number 1247918