On a normal integral bases problem over cyclotomic \(\mathbb{Z}_p\)-extensions
type: Item
instance of: scholarly article
title: On a normal integral bases problem over cyclotomic \(\mathbb{Z}_p\)-extensions
zbMATH Open document ID: 0892.11036
author: Humio Ichimura
published in: Journal of the Mathematical Society of Japan
publication date: 1997-08-31T00:00:00Z
review text: This paper studies the Kummer extensions \(K(\alpha^{(1/p)})\) of a number field which are unramified and possess a relative normal basis (RNB). The main topic is the study of the situation where \(K\) varies amongst the levels of a cyclotomic \(\mathbb{Z}_p\) extension. Using the complex conjugation, one can break up the problem into two parts: \textit{J. Brinkhuis} [Manuscr. Math. 75, 333-347 (1992; Zbl 0757.11044)] has already proved that the minus part is trivial. So this paper concentrates on the \(\psi\)-part with \(\psi\) an even character. Theorem 2 says that if for every \(\alpha\) (in the \(\psi\)-part), the corresponding extension, if unramified, has a RNB, then the Iwasawa \(\Lambda\)-invariant vanishes. Theorem 3 gives a criterion in terms of a congruence modulo \(p\) of special values of \(L\)-functions. Theorem 4 is a kind of capitulation result: every unramified cyclic extension of degree \(p\) over \(k_n\) gets a RNB when passing over \(k_{n+1}\). Theorem 1 gives the application to the existence of a RNB when \(p\) does not divide the + part of the class number.
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R33
zbMATH DE Number: 1050410
zbMATH Keywords: Kummer extensions
zbMATH Keywords: relative normal basis
zbMATH Keywords: cyclotomic \(\mathbb{Z}_p\) extension
zbMATH Keywords: Iwasawa \(\Lambda\)-invariant
zbMATH Keywords: special values of \(L\)-functions
reviewed by: Roland Gillard
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04840689
OpenAlex ID: W2074653705
DOI: 10.2969/JMSJ/04840689
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description: scientific article; zbMATH DE number 1050410