A remark on Boston's question concerning the existence of unramified \(p\)-extensions
type: Item
instance of: scholarly article
title: A remark on Boston's question concerning the existence of unramified \(p\)-extensions
zbMATH Open document ID: 0856.11052
author: Akito Nomura
published in: Journal of Number Theory
publication date: 1996-07-09T00:00:00Z
review text: Let \(K\) be a number field, \(p\) an odd prime, \(K(p)\) the Hilbert \(p\)-class field of \(K\), and assume that the class number of \(K(p)\) is divisible by \(p\). \textit{N. Boston} [J. Number Theory 42, 285-291 (1992; Zbl 0768.11044)] asked whether there always exists a nontrivial unramified extension \(M/K\) such that \(\text{Gal}(M/K)\) is a \(p\)-group with the same exponent as \(\text{Gal} (K(p)/K)\), and remarked that the truth of the Fontaine-Mazur conjecture implies an affirmative answer if \(K\) has an infinite \(p\)-class field tower. Although such an extension \(M\) does not always exist [see \textit{N. Boston}, Some cases of the Fontaine-Mazur conjecture. II, J. Number Theory (to appear)], the author provides some support for the existence of \(M\) in the semisimple case by proving the following theorem, using the methods he developed in [Manuscr. Math. 79, 379-390 (1993; Zbl 0806.11053)]: Let \(p\) and \(\ell\) be distinct odd primes, and let \(k\) be a number field with finite unit group and class number prime to \(p\). Assume that \(K\) is an abelian \(\ell\)-extension such that \(p\) divides the class number of \(K\). If the order \(f\) of \(p\) modulo \(\ell\) is even, then the class number of \(K(p)\) is also divisible by \(p\), and there exists an unramified non-abelian \(p\)-extension \(M/K\) with exponent \(p\).
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R20
zbMATH DE Number: 883834
zbMATH Keywords: Hilbert class field
zbMATH Keywords: Fontaine-Mazur conjecture
zbMATH Keywords: \(p\)-class field tower
zbMATH Keywords: \(p\)-extension
MaRDI profile type: Publication
full work available at URL: jnth.1996.0063
OpenAlex ID: W2036859512
DOI: 10.1006/JNTH.1996.0063
description: scientific article; zbMATH DE number 883834