On the Galois module structure of cyclic Kummer extensions
type: Item
instance of: scholarly article
title: On the Galois module structure of cyclic Kummer extensions
zbMATH Open document ID: 1179.11038
author: Yoshimasa Miyata
published in: Journal of Algebra
publication date: 2008-12-05T00:00:00Z
review text: Let \(K/k\) be a finite Galois extension of fields with Galois group \(G\). Then the Normal Basis Theorem says that \(K\) is a free rank \(1\) module over the group algebra \(k[G]\). Now suppose that \(K\) and \(k\) are either number fields or \(p\)-adic fields, and let \(\mathfrak{O}_{K}\) and \(\mathfrak{O}_{k}\) respectively denote their rings of integers. Let \(\mathfrak{A} := \{ \lambda \in k[G] \mid \lambda \mathfrak{O}_{K} \subseteq \mathfrak{O}_{K} \}\) be the associated order. Then a natural problem is to determine \(\mathfrak{A}\) and the structure of \(\mathfrak{O}_{K}\) as a (left) \(\mathfrak{A}\)-module. When \(K/k\) is (at most) tamely ramified, \(\mathfrak{O}_{K}\) is free (resp.\ locally free) over \(\mathfrak{A} = \mathfrak{O}_{k}[G] \) in the \(p\)-adic case (resp.\ number field case). However, the situation is somewhat more complicated when \(K/k\) is wildly ramified; in particular \(\mathfrak{O}_{k}[G] \subsetneq \mathfrak{A}\). The first part of the paper under review considers situation in which \(K/k\) is a totally and wildly ramified cyclic Kummer extension of \(p\)-adic fields of odd prime power degree \(p^{n}\). Suppose that \(K=k(\alpha)\) where \(\alpha^{p^{n}} \in \mathfrak{O}_{k}\) and \((\mathrm{val}_{K}(\alpha-1),p)=1\). In [\textit{Y.\ Miyata}, Math. Proc. Camb. Philos. Soc. 123, No. 2, 199--212 (1998; Zbl 1073.11525)], the author gave a necessary and sufficient condition for \(\mathfrak{O}_{K}\) to be free (necessarily of rank \(1\)) over \(\mathfrak{A}\). Note that every intermediate field of \(K/k\) is of the form \(K_{l} := k(\alpha^{p^{n-l}})\) with \(0 \leq l \leq n\). Let \(\mathfrak{O}_{l}\) denote the ring of integers of \(K_{l}\) and let \(\mathfrak{A}_{l}\) be the associated order of \(K_{l}/k\) contained in the group algebra \(k[\mathrm{Gal}(K_{l}/k)]\). Theorem 4 of the paper under review gives a condition for \(\mathfrak{O}_{l}\) to be free over \(\mathfrak{A}_{l}\) for \textit{every} \(l\). This condition involves the…
reviewed by: Henri Johnston
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11S23
zbMATH DE Number: 5378812
zbMATH Keywords: Galois module structure
zbMATH Keywords: local field
zbMATH Keywords: free module
zbMATH Keywords: isomorphism class
zbMATH Keywords: Stickelberger ideal
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.06.034
OpenAlex ID: W2036244368
cites work: Q5644060
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cites work: Q3735800
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cites work: Wildly Ramified Gamma Extensions
DOI: 10.1016/J.JALGEBRA.2008.06.034
description: scientific article; zbMATH DE number 5378812