A note on free pro-\(p\)-extensions of algebraic number fields. II
type: Item
instance of: scholarly article
title: A note on free pro-\(p\)-extensions of algebraic number fields. II
zbMATH Open document ID: 0869.11086
DOI: 10.1007/BF02567952
author: Masakazu Yamagishi
published in: Manuscripta Mathematica
publication date: 1997-09-01T00:00:00Z
full work available at URL: https://eudml.org/doc/156227
review text: For a rational prime \(p\) and an algebraic number field \(k\), a Galois extension \(K/k\) whose Galois group is isomorphic to the free pro-\(p\)-group on \(d\) generators will be called a free pro-\(p\)-extension of rank \(d\), or an \(F_d\)-extension. Let \(\rho= \rho(k)\) be the maximal rank of all free pro-\(p\)-extensions of \(k\). Leopoldt's conjecture immediately implies that \(\rho\leq 1+r_2\), where \(r_2\) is the number of complex places of \(k\). But examples exist for which \(\rho<1 +r_2\) [see Part I, J. Théor. Nombres Bordx. 5, 165-178 (1993; Zbl 0784.11052)]. Besides, an \(F_\rho\)-extension is not necessarily unique. In this short note, improving upon an observation of Iwasawa, the author shows that if \(k\) admits an \(F_{1+r_2}\)-extension \(K\) which contains the cyclotomic \(\mathbb{Z}\)-extension of \(k\), then \(K\) is unique.
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 980692
zbMATH Keywords: free pro-\(p\)-group
zbMATH Keywords: free pro-\(p\)-extension
zbMATH Keywords: cyclotomic extension
zbMATH Keywords: Galois extension
zbMATH Keywords: maximal rank
reviewed by: Thong Nguyen Quang Do
MaRDI profile type: Publication
cites work: On \(\mathbb Z_{\ell}\)-extensions of algebraic number fields
cites work: On the maximal unramified p-extension of an algebraic number field.
cites work: A note on free pro-\(p\)-extensions of algebraic number fields
Wikidata QID: Q126271038
OpenAlex ID: W3129013637
description: scientific article; zbMATH DE number 980692