On the pro-\(p\)-extensions with restricted ramification over the cyclotomic \(\mathbb Z_p\)-extension of a number field
type: Item
instance of: scholarly article
title: On the pro-\(p\)-extensions with restricted ramification over the cyclotomic \(\mathbb Z_p\)-extension of a number field
zbMATH Open document ID: 1163.11071
author: Landry Salle
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2009-04-14T00:00:00Z
full work available at URL: https://eudml.org/doc/10847
review text: This article, first part of the author's thesis [\textit{L. Salle}, Présentations de groupes de Galois de pro-\(p\)-extensions de corps de nombres, Thèse de l'Université Toulouse III (2008)], deals with the maximal pro-\(p\)-extensions with restricted ramification over the cyclotomic \({\mathbb Z}_p\)-extension over a number field. The author investigates their Galois group, focusing first on the \({\mathbb Z}_p\)-rank of their abelianization and then on their minimal numbers of generators and relations. The author uses class field theory and Koch Galois theory of pro-\(p\)-extensions in [\textit{H. Koch}, Galois theory of \(p\)-extensions. Berlin: Springer (2002; Zbl 1023.11002)]. Let \(k\) a field of algebraic numbers, \(p\) a prime, \(k_\infty\) the cyclotomic \({\mathbb Z}_p\)-extension of \(k\), and \(S\) a finite set of places of \(k\). In this article the author examines the maximal \(S\)-ramified pro-\(p\)-extension \(\overline{L}_S(k_\infty)\) of \(k_\infty\) and the maximal abelian sub-extension \(\overline{L}_S(k_\infty)^{ab}\) of \(\overline{L}_S(k_\infty)/k\). When \(S=\emptyset\), the corresponding maximal abelian extension, denoted \(\overline{L}(k_\infty)^{ab}\) is the cyclotomically ramified extension, following the article [\textit{J.-F. Jaulent} and \textit{J. Sands}, Compos. Math. 99, No. 3, 325--341 (1995; Zbl 0869.11084)]. In the other hand, when \(S\) contains all the places over \(p\), this extension is precisely the maximal \(S\)-ramified pro-\(p\)-extension. In a first part, the author considers the \({\mathbb Z}_p\)-free part of the group \(\text{Gal}(\overline{L}_S(k_\infty)^{ab}/k)\) caracterized as a \({\mathbb Z}_p\)-module by its \({\mathbb Z}_p\)-rank. Several results exists on this \({\mathbb Z}_p\)-rank in Jaulent and Sands article [loc. cit.] when \(S=\emptyset\). This article systematizes this approach when \(S\not=\emptyset\) with the use of the \(p\)-adic theory as developed by \textit{J.-F. Jaulent} in [J. Théor. Nombres Bordx…
reviewed by: Roland Quême
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11S15
Mathematics Subject Classification ID: 11S20
Mathematics Subject Classification ID: 11S25
zbMATH DE Number: 5543173
zbMATH Keywords: Iwasawa theory
zbMATH Keywords: Galois theory of \(p\)-extensions
zbMATH Keywords: \(p\)-adic class field theory
zbMATH Keywords: Galois cohomology
MaRDI profile type: Publication
OpenAlex ID: W1988725363
cites work: Q4779057
cites work: Global \(\ell\)-adique clean field theory
cites work: Sur les invariants d'Iwasawa des tours cyclotomiques
cites work: Q4863732
cites work: Q4553271
cites work: Local extensions associated with \(\ell\)-extensions of number fields with bounded ramification
cites work: Q4935965
cites work: Q4193645
DOI: 10.5802/JTNB.638
description: scientific article; zbMATH DE number 5543173