A note on free pro-\(p\)-extensions of algebraic number fields
type: Item
instance of: scholarly article
title: A note on free pro-\(p\)-extensions of algebraic number fields
zbMATH Open document ID: 0784.11052
author: Masakazu Yamagishi
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1994-01-26T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93570
review text: The author studies the maximal rank \(\rho\) of free pro-\(p\) Galois groups \(F_ \rho\) over a fixed algebraic number field \(k\) for a fixed prime \(p\). Such an \(F_ \rho\) is necessarily a quotient of \(G_{S_ p}\), the Galois group over \(k\) of the maximal pro-\(p\)-extension of \(k\) unramified outside \(p\). By class field theory, the rank of \(G_{S_ p}^{ab}\) is \(1+r_ 2+\delta\), where \(r_ 2\) denotes the number of complex places of \(k\), and \(\delta\) the defect of Leopoldt's conjecture. The main results are the following: 1) \(\rho\leq 1+r_ 2\) if the ``weak Leopoldt conjecture'' holds for all \(\mathbb{Z}_ p\)-extensions of \(k\). 2) There exist \(k\), \(p\) such that \(\rho<1+r_ 2\). Such examples come from number fields \(k\) for which \(G_{S_ p}\), under rather strong conditions, is a quotient of a free pro-\(p\)-product of decomposition groups and a free group [see \textit{K. Wingberg}, J. Reine Angew. Math. 400, 185-202 (1989; Zbl 0715.11065)].
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R34
zbMATH DE Number: 484169
zbMATH Keywords: weak Leopoldt conjecture
zbMATH Keywords: maximal rank
zbMATH Keywords: free pro-\(p\) Galois groups
reviewed by: Thong Nguyen Quang Do
MaRDI profile type: Publication
OpenAlex ID: W2010081823
cites work: ON SOME QUESTIONS IN THE THEORY OF Γ-EXTENSIONS OF ALGEBRAIC NUMBER FIELDS
cites work: On regular number fields
cites work: On the structure of certain Galois groups
cites work: On \(\mathbb Z_{\ell}\)-extensions of algebraic number fields
cites work: Q4710489
cites work: LOCAL EXTENSIONS ASSOCIATED WITH $ l$-EXTENSIONS WITH GIVEN RAMIFICATION
cites work: Classification of Demushkin Groups
cites work: Sur les p -extensions des corps p -rationnels
cites work: Q3481807
cites work: Freie Produkte pro-endlicher Gruppen und ihre Kohomologie
cites work: Q3937499
cites work: Q3331297
cites work: Q4205513
cites work: Extensions with given points of ramification.
cites work: Galois Cohomology
cites work: Epimorphisms of Demushkin groups
cites work: Q3926705
cites work: Freie Produktzerlegungen von Galoisgruppen und Iwasawa-Invarianten für p-Erweiterungen von
cites work: Q3748371
cites work: On Galois groups of p-closed algebraic number fields with restricted ramification.
cites work: On Galois groups of p-closed algebraic number fields with restricted ramification II.
cites work: On the center of Galois groups of maximal pro-p extensions of algebraic number fields with restricted ramification.
DOI: 10.5802/JTNB.86
description: scientific article; zbMATH DE number 484169