Relative Galois module structure of integers of abelian fields
type: Item
instance of: scholarly article
title: Relative Galois module structure of integers of abelian fields
zbMATH Open document ID: 0859.11059
author: Nigel P. Byott
author: Günter Lettl
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1997-04-09T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/228275
full work available at URL: jtnb8-1.html
review text: The paper explicitly describes the associated order \({\mathfrak A}_{L/K}\) of a number field extension \(L/K\) with \(L/\mathbb Q\) abelian and \(K\) a cyclotomic field. As a first corollary it is shown that \({\mathfrak A}_{L/K}\) is isomorphic to the ring of integers in \(L\). Special cases have been studied before by \textit{H. W. Leopoldt} [J. Reine Angew. Math. 201, 119--149 (1959; Zbl 0098.03403)] where \(K=\mathbb Q\), by \textit{S.-P. Chan} and \textit{C.-H. Lin} [J. Reine Angew. Math. 434, 205--220 (1993; Zbl 0753.11038)] where \(L\) is cyclotomic, and by \textit{W. Bley} [Can. Math. Bull. 38, 141--148 (1995; Zbl 0830.11037)] where \(\mathbb Q(\zeta_n)/L\) is tame and \(n\) is the conductor of \(L\). A second corollary says that, for more general subfields \(K\) of \(L\), \({\mathfrak A}_{L/K}\) is the maximal order in \(K[\text{Gal}(L/L_m)]\) provided that \(L/K\) is cyclic, totally wildly ramified, and disjoint to \(\mathbb Q(\zeta_{m'})/K\). Here, \(m'=\) conductor of \(K\), \(m=m' \cdot \prod p\) with \(p\) running through the prime divisors of \(n\) which do not divide \(m'\), and \(L=L\cap \mathbb Q(\root m\of {1})\).
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R20
zbMATH DE Number: 912502
zbMATH Keywords: relative Galois module structure
zbMATH Keywords: abelian extension of a cyclotomic field
zbMATH Keywords: order
reviewed by: Jürgen Ritter
MaRDI profile type: Publication
OpenAlex ID: W1980025704
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DOI: 10.5802/JTNB.160
description: scientific article; zbMATH DE number 912502