Cyclotomic units in \(\mathbb{Z}_p\)-extensions
type: Item
instance of: scholarly article
title: Cyclotomic units in \(\mathbb{Z}_p\)-extensions
zbMATH Open document ID: 0765.11042
DOI: 10.1007/BF02776022
author: Jae Moon Kim
author: Sunghan Bae
author: In-Sok Lee
published in: Israel Journal of Mathematics
publication date: 1992-12-16T00:00:00Z
review text: Let \(K_ n=\mathbb{Q}(\zeta_{p^{n+1}}+ \zeta_{p^{n+1}}^{-1})\), where \(p\) is an odd prime and \(\zeta_{p^k}\) is a primitive \(p^k\)-th root of unity. Let \(A_n\) denote the \(p\)-part of the ideal class group of \(K_n\), and \(B_n\) the \(p\)-part of the factor group of the cyclotomic units in the unit group of \(K_n\). The authors describe the structure of the direct and inverse limits of \(B_n\). In particular, \(\varinjlim B_n\simeq (\mathbb{Q}_p/\mathbb{Z}_p)^\lambda \oplus M\), where \(\lambda\) is the Iwasawa invariant and \(M\) is a finite group. It is proved that \(M\simeq C\) where \(C\) is determined by \(\varprojlim A_n\simeq \mathbb{Z}_p^\lambda \oplus C\).
reviewed by: Veikko Ennola
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 78180
zbMATH Keywords: direct limit
zbMATH Keywords: ideal class group
zbMATH Keywords: \(p\)-part
zbMATH Keywords: factor group
zbMATH Keywords: cyclotomic units
zbMATH Keywords: inverse limits
zbMATH Keywords: Iwasawa invariant
MaRDI profile type: Publication
cites work: Q4205511
cites work: Sur la capitulation dans une Zl-extension.
cites work: On \(\mathbb Z_{\ell}\)-extensions of algebraic number fields
cites work: Cohomology groups of cyclotomic units
cites work: Q3856855
full work available at URL: https://doi.org/10.1007/bf02776022
OpenAlex ID: W2066502793
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description: scientific article; zbMATH DE number 78180