Finiteness of \(T\)-tamely ramified and \(S\)-decomposed towers and \(p\)-towers
type: Item
instance of: scholarly article
title: Finiteness of \(T\)-tamely ramified and \(S\)-decomposed towers and \(p\)-towers
zbMATH Open document ID: 0877.11060
author: Christian Maire
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1996-11-07T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/230944
full work available at URL: jtnb8-1.html
review text: If \(T\) and \(S\) are (finite, disjoint) sets of places of a number field \(k\), let \(k_1= k_1^{S,T}\) be the greatest abelian extension of \(k\), unramified outside \(T\), moderately ramified on \(T\), where places of \(S\) split completely. Denoting by \(S_1\) and \(T_1\) the places of \(k_1\) lying over \(S\) and \(T\), one can iterate this construction, obtaining the \(T-S\)-hilbert tower of \(k\). The author begins with a finiteness criterion of the \(p\)-part of this tower (for fixed prime number \(p)\). Namely, if the \(p\)-Sylow such group of the Galois group \(\text{Gal} (k_1/k) = cl^{S,T} (k)\) is cyclic, then the tower is finite (and in fact stops at the first stage). Then, the author gives two nonfiniteness criteria of the tower (as usual, both involving the Golod-Shafarevich theorem on finite \(p\)-groups), the second one involving Jaulent's genus theory. Many examples are given. For instance, the field \(\mathbb{Q}(\sqrt{53.131})\) has infinite tower in the narrow sense (i.e. for \(T\) and \(S\) empty), but finite tower in the classical sense (i.e., \(T\) and \(S\) empty), \(S=S^\infty)\). The author concludes with considerations (using Odlyzko bounds) on the discriminant of fields having infinite tower.
reviewed by: Marc Perret
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 912498
zbMATH Keywords: Galois theory
zbMATH Keywords: number fields
zbMATH Keywords: genus theory
zbMATH Keywords: Hilbert tower
zbMATH Keywords: finiteness criterion
zbMATH Keywords: discriminant
MaRDI profile type: Publication
OpenAlex ID: W2324104825
cites work: Q5094815
cites work: Q3735801
cites work: Number theory II: Algebraic number theory. Transl. from the Russian
cites work: Tours de corps de classes et estimations de discriminants
cites work: ON THE CLASS FIELD TOWER OF AN IMAGINARY QUADRATIC NUMBER FIELD
cites work: Tours ramifiées infinies de corps de classes. (Infinite ramified class field towers)
cites work: Q5094822
cites work: Konstruktion imaginärquadratischer Körper mit unendlichem Klassenkörperturm
cites work: Infinite class field towers of quadratic fields.
cites work: Q3400910
DOI: 10.5802/JTNB.156
description: scientific article; zbMATH DE number 912498