On the growth of \(p\)-class groups in \(p\)-class field towers
type: Item
instance of: scholarly article
title: On the growth of \(p\)-class groups in \(p\)-class field towers
zbMATH Open document ID: 0879.11069
author: Farshid Hajir
published in: Journal of Algebra
publication date: 1998-01-18T00:00:00Z
review text: Let \(k\) be an algebraic number field, \(p\) a prime and \(G\) the Galois group of the maximal unramified \(p\)-extension of \(k\). Furthermore, let \(\{G^{(n)} \mid n=0,1, \dots\}\) be the derived series of \(G\) such that \(G^{(n+1)}\) is the Hilbert \(p\)-class field of the fixed field \(k^{(n)}\) of \(G^{(n)}\). The author investigates the question of whether the generator rank \(d_n\) of \(G^{(n)}\) tends to infinity with \(n\). He proves that this is the case if \(d_0\geq 2+2 (\varepsilon +1)^{1/2}\), where \(\varepsilon\) is the \(p\)-rank of the group of units of \(k\). He also shows that a conjecture of Fontaine and Mazur implies \(\lim_{n \to\infty} d_n= \infty\) if and only if \(G\) is infinite. For other results of the author we refer to the paper at hand and his related paper [Pac. J. Math. 176, 15-18 (1996; see Zbl 0879.11068 above)].
Mathematics Subject Classification ID: 11S31
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 1013923
zbMATH Keywords: \(p\)-class groups
zbMATH Keywords: \(p\)-class field towers
zbMATH Keywords: Hilbert \(p\)-class field
reviewed by: Helmut Koch
MaRDI profile type: Publication
full work available at URL: jabr.1996.6849
OpenAlex ID: W2050121250
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DOI: 10.1006/JABR.1996.6849
description: scientific article; zbMATH DE number 1013923