Circular units in the \(\mathbb Z_p\)-extensions of real abelian fields of prime conductor
type: Item
instance of: scholarly article
title: Circular units in the \(\mathbb Z_p\)-extensions of real abelian fields of prime conductor
zbMATH Open document ID: 0948.11039
author: Jae Moon Kim
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2000-11-20T00:00:00Z
review text: Let \(k\) be a real abelian field and \(k_\infty\) its cyclotomic \(\mathbb Z_p\)-extension, \(p\) being a prime prime to the degree \([k:\mathbb Q]\). Let \(G_{m,n}\) be the Galois group between two different layers \(k_n\) and \(k_m\) in \(k_\infty (n\leq m)\). The paper computes the Galois cohomology of the Sinnott group of cyclotomic units \(C_m\) in \(k_m\): 1) \(H^0(G_{m,n},C_m)=C_n\), 2) \(\widehat H^0(G_{m,n},C_m)\simeq (\mathbb Z_p/p^{m-n}\mathbb Z_p)^{l-1}\), 3) \(\widehat H^1(G_{m,n},C_m)\simeq (\mathbb Z_p/p^{m-n}\mathbb Z_p)^{l}\) The main tool is \textit{V. Ennola}'s work on relations of cyclotomic units [J. Number Theory 4, 236--247 (1972; Zbl 0236.12005)].
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R33
zbMATH DE Number: 1389722
zbMATH Keywords: Galois cohomology
zbMATH Keywords: Sinnott group
zbMATH Keywords: cyclotomic units
reviewed by: Roland Gillard
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178224765
OpenAlex ID: W2035734874
cites work: On relations between cyclotomic units
cites work: On Cohomology Groups of Units for Z p -Extensions
cites work: Units and Cyclotomic Units in Zp-Extensions
cites work: Class numbers of real quadratic fields
cites work: Q5612629
cites work: On the Stickelberger ideal and the circular units of an abelian field
cites work: Q5690489
DOI: 10.2748/TMJ/1178224765
description: scientific article; zbMATH DE number 1389722