On Galois structure of the integers in elementary abelian extensions of local number fields
type: Item
instance of: scholarly article
title: On Galois structure of the integers in elementary abelian extensions of local number fields
zbMATH Open document ID: 1185.11068
author: Yoshimasa Miyata
published in: Journal of Number Theory
publication date: 2007-09-13T00:00:00Z
review text: Let \(p\) be an odd prime number and \(G\) an elementary abelian group of order \(p^2\). Let \(k\) be a finite extension of \(\mathbb{Q}_p\) containing a primitive \(p\)th root of unity. Let \(O_k\) be the ring of integers of \(k\). Let \(K/k\) be a totally ramified Galois (Kummer) extension with Galois group \(G\), and \(O_K\) its ring of integers. Let \(K'/k\) be another such extension. The author gives a necessary and sufficient condition for \(O_K\) and \(O_{K'}\) to be \(O_k[G]\)-isomorphic. From this result he deduces that there exist \(K/k\) and \(K'/k\) such that \(O_K\) and \(O_{K'}\) are \(\mathbb{Z}_p[G]\)-isomorphic, but not \(O_k[G]\)-isomorphic. In the last section, the author obtains sufficient condition, in term of ramification, for \(O_K\) to be not free over its associated order in \(k[G]\).
reviewed by: Bouchaïb Sodaïgui
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11S20
zbMATH DE Number: 5190752
zbMATH Keywords: Kummer extensions over local fields
zbMATH Keywords: isomorphism class of rings of integers
zbMATH Keywords: free modules over associated order
MaRDI profile type: Publication
full work available at URL: j.jnt.2006.12.005
OpenAlex ID: W2058729841
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DOI: 10.1016/J.JNT.2006.12.005
description: scientific article; zbMATH DE number 5190752