On regular number fields
type: Item
instance of: scholarly article
title: On regular number fields
zbMATH Open document ID: 0704.11040
DOI: 10.1007/BF01159964
author: Georges Gras
author: Jean-François Jaulent
published in: Mathematische Zeitschrift
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174095
review text: Fix a prime number \(\ell\) and let \(\zeta\) denote a primitive \(\ell\)-th root of unity. Let \(K\) be a number field with divisor group \(D_ K\) and write \(\mathcal D_ K\) for the multiplicative tensor product \(\mathbb Z_{\ell}\otimes_{\mathbb Z}D_ K\). One can define a surjective map (called Gras's logarithm), \(\text{lg}: \mathcal D_ K\to \text{Gal}(Z/K)\), where \(Z\) is the composite field of the \(\mathbb Z_{\ell}\)-extensions of \(K\). For a finite or a real place \(\wp\) of \(K\), \(\text{lg}(\wp)=1\) whenever \(\wp\) is real or lies over \(\ell\), otherwise \(\text{lg}(\wp)\) is a topological generator of the decomposition group \(D_{\wp}(Z/K)\simeq\mathbb Z_{\ell}\) associated to \(\wp\) in the abelian extension \(Z/K\). A finite set \(S\) of places of \(K\) is called primitive when the \(\text{lg}(s)\), \(s\in S\), form a \(\mathbb Z_{\ell}\)-basis of a pure submodule of \(\text{Gal}(Z/K)\). An \(\ell\)-extension \(L/K\) is called primitively ramified if the set \(S\) of places of \(K\) that ramify tamely in \(L/K\), is primitive. \(K\) is called regular (with respect to \(\ell\)) if the \(\ell\)-Sylow subgroup \(R_ 2(K)\) of the kernel in \(K_ 2(K)\) of the regular symbols attached to the non-complex places of \(K\), is trivial. When \(K\) contains the maximal real subfield \(k=\mathbb Q(\zeta +\zeta^{-1})\) of the cyclotomic field \(\mathbb Q(\zeta)\) several equivalent characterizations of regularity can be given, one of which says that \(K\) is regular if and only if \(K\) verifies Leopoldt's conjecture (with respect to \(\ell)\) and the torsion submodule \({\mathcal T}_ K\) of \(\text{Gal}(M/K)\), with \(M\) the maximal \(\ell\)-ramified, abelian \(\ell\)-extension of \(K\) which decomposes completely at the infinite places, is zero. A classical example of a regular number field is provided by the cyclotomic field \(\mathbb Q(\zeta)\) if and only if \(\ell\) is a regular prime in the usual terminology. Writing \(\delta_ K\) for the defect of Leop…
reviewed by: W. W. J. Hulsbergen
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R70
Mathematics Subject Classification ID: 11S15
Mathematics Subject Classification ID: 19C99
Mathematics Subject Classification ID: 19F15
zbMATH DE Number: 4154554
zbMATH Keywords: Leopoldt's conjecture
zbMATH Keywords: regular number field
zbMATH Keywords: \(\ell \)-rational field
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4154554