Vertices of ideals of a \({\mathfrak p}\)-adic number field
type: Item
instance of: scholarly article
title: Vertices of ideals of a \({\mathfrak p}\)-adic number field
zbMATH Open document ID: 0611.12013
author: Yoshimasa Miyata
published in: Illinois Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Let \(K/k\) be a finite normal extension with Galois group \(G\) over a \({\mathfrak p}\)-adic number field, and \(\pi\) be a prime element of the ring of integers in \(K\). The author intends to study the structure of ideals \((\pi^ i)\) of \(K\) as Galois module, and first defines ''the vertex \(V(\pi^ i)\) of an ideal \((\pi^ i)\)'' as the minimal normal subgroup \(S\) of \(G\) such that \((\pi^ i)\) is \((G,S)\)-projective, i.e. relatively projective with respect to a subgroup \(S\) of \(G\). He provides explicitly some fundamental relations between the vertex \(V(\pi^ i)\) and the j-th ramification group \(G_ j\) of \(K/k\). One of them is \(G_ 1\supseteq V(\pi^ i)\supseteq G_ 2\) for any i, which is a generalization of \textit{A. Fröhlich}'s result [cf. Algebr. Zahlentheorie, Ber. Tagung math. Forschinst. Oberwolfach 1964, 59-82 (1966; Zbl 0199.097)].
reviewed by: Hideo Yokoi
Mathematics Subject Classification ID: 11S15
Mathematics Subject Classification ID: 11S20
Mathematics Subject Classification ID: 13C10
zbMATH DE Number: 3987402
zbMATH Keywords: Galois module structure
zbMATH Keywords: vertex of ideal
zbMATH Keywords: \({\mathfrak p}\)-adic number field
zbMATH Keywords: relatively projective
zbMATH Keywords: ramification group
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3987402