On Iwasawa \(\lambda_p\)-invariants of relative real cyclic extensions of degree \(p\)
type: Item
instance of: scholarly article
title: On Iwasawa \(\lambda_p\)-invariants of relative real cyclic extensions of degree \(p\)
zbMATH Open document ID: 0919.11068
author: Takashi Fukuda
author: Keiichi Komatsu
author: Manabu Ozaki
author: Hisao Taya
published in: Tokyo Journal of Mathematics
publication date: 1999-05-30T00:00:00Z
review text: Let \(K\) be a totally real number field whose class number is not divisible by \(p\), and assume that \(L/K\) is a real cyclic extension of prime degree \(p\). Let \(K_\infty\) denote the cyclotomic \(\mathbb{Z}_p\)-extension of \(K\) and assume that there is only one prime in \(K_\infty\) above \(p\). The authors prove the following condition for the vanishing of the \(\lambda\)-invariant of \(L\): we have \(\lambda_p(L)=0\) if and only if the prime ideals \({\mathfrak p}_i\) in \(L_\infty\) that are prime to \(p\) and ramified in \(L_\infty/K_\infty\) generate ideal classes in \(L_\infty\) whose orders are not divisible by \(p\). Applications are given to the special case of such cyclic cubic extensions of \(K=\mathbb{Q}\) for which the splitting fields of the \({\mathfrak p}_i\) are small.
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 1130811
zbMATH Keywords: Iwasawa theory
zbMATH Keywords: \(\lambda\)-invariants
zbMATH Keywords: Greenberg's conjecture
zbMATH Keywords: real cyclic extension
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270042120
OpenAlex ID: W2090352589
DOI: 10.3836/TJM/1270042120
description: scientific article; zbMATH DE number 1130811