\(p\)-ranks of class groups of Witt equivalent number fields
type: Item
instance of: scholarly article
title: \(p\)-ranks of class groups of Witt equivalent number fields
zbMATH Open document ID: 0988.11054
published in: Journal of Number Theory
publication date: 2002-02-21T00:00:00Z
review text: Two algebraic number fields are said to be Witt equivalent if their Witt rings of symmetric bilinear forms are isomorphic (a necessary condition for two fields to be Witt equivalent is that their field degrees over \(\mathbb{Q}\) are equal). Let \(n\) be a prescribed positive integer. In this paper it is proved that, for each prime \(p\) dividing \(n\), every infinite class of Witt equivalent number fields of degree \(n\) contains a field for which the \(p\)-rank of the ideal class group exceeds any given bound. This generalizes the class for \(p=2\) and fields of even degree, for which the result was established previously by the author [Ann. Math. Sil. 12, 53-64 (1998; Zbl 0926.11083)]. Reformulated in terms of Hilbert class fields, the result asserts the existence of a field with infinite Hilbert \(p\)-class field tower in every infinite class of Witt equivalent number fields of degree \(n\). The existence of a field with the properties claimed in the main theorem is established using a theorem of \textit{P. Roquette} and \textit{H. Zassenhaus} [J. Lond. Math. Soc. 44, 31-38 (1969; Zbl 0169.38001)].
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 1347932
zbMATH Keywords: \(p\)-rank of ideal class group
zbMATH Keywords: Witt equivalent number fields
zbMATH Keywords: Hilbert class fields
zbMATH Keywords: infinite Hilbert \(p\)-class field tower
author: Kazimierz Szymiczek
reviewed by: Andrew G. Earnest
MaRDI profile type: Publication
full work available at URL: jnth.1999.2397
OpenAlex ID: W2000664963
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DOI: 10.1006/JNTH.1999.2397
description: scientific article; zbMATH DE number 1347932