Normal integral bases of \(\infty\)-ramified abelian extensions of totally real number fields.
type: Item
instance of: scholarly article
title: Normal integral bases of \(\infty\)-ramified abelian extensions of totally real number fields.
zbMATH Open document ID: 1036.11058
DOI: 10.1007/BF02941673
author: Fuminori Kawamoto
author: Yoshitaka Odai
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2003-02-13T00:00:00Z
review text: Let \(F\) be a totally real algebraic number field, \(\mathfrak l\) a product of (some) real primes of \(F\) and \(K\) an abelian extension of \(F\) with conductor \(\mathfrak l\). \textit{J. Brinkhuis} [Bull. Lond. Math. Soc. 24, 236--242 (1992; Zbl 0768.11045)] proved that if \(K/F\) has a normal integral basis then \(\text{Gal}(K/F)\) must be an elementary 2-group. In the present paper, the authors construct a maximal extension field \(L^{\mathfrak l}\) of \(F\) such that all abelian extensions \(K\) of \(F\) with conductor dividing \(\mathfrak l\) and having a normal integral basis over \(F\) are exactly the fields between \(F\) and \(L^{\mathfrak l}\). The proof uses Kummer theory and a result of \textit{L. N. Childs} [Proc. Lond. Math. Soc. (3) 35, 407--422 (1977; Zbl 0374.13002)] to characterize the existence of normal integral bases for quadratic extensions unramified outside \(\infty\). The larger part of the paper gives a detailed and explicit description of the situation where \(F\) is either a real quadratic or a cyclic cubic number field.
reviewed by: Günter Lettl
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11R80
zbMATH DE Number: 1868556
zbMATH Keywords: ray class field
zbMATH Keywords: Kummer theory
zbMATH Keywords: elementary 2-group
describes a project that uses: UBASIC
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02941673
OpenAlex ID: W1965871363
description: scientific article; zbMATH DE number 1868556