Galois structure of ideals in wildly ramified abelian \(p\)-extensions of a \(p\)-adic field, and some applications
type: Item
instance of: scholarly article
title: Galois structure of ideals in wildly ramified abelian \(p\)-extensions of a \(p\)-adic field, and some applications
zbMATH Open document ID: 0889.11040
author: Nigel P. Byott
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1998-03-25T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93679
full work available at URL: jtnb9-1.html
review text: It is shown that fairly frequently ideals in the ring of integers of on extension \(L\) of \(K\) fail to be free over their associated order. Here \(K\) is a \(p\)-adic field and \(L/K\) is a finite abelian \(p\)-extension with ramification index \(p^n\). An efficient criterion for an ideal \(I\) not to be free over its associated order \(\mathcal A\) is given in terms of ramification numbers: if \(I\) is free, then, under a very mild extra condition which we don't spell out here, all ramification numbers of \(L/K\) have to be congruent to \(-1\) modulo \(p^n\). The first step towards this result is a very simple, yet clever statement (Theorem 2.1), which is worth being quoted in full: if a module \(M\) over the local ring \(\mathcal A\) has a system of generators \(m_1,\ldots,m_n\) such that none of the \(m_i\) is a generator individually, then \(M\) is not free of rank one. One other main source is \textit{S. Vostokov}'s criterion for an ideal \(I\) to be indecomposable as a module over \({\mathcal O}_K[\)Gal\((L/K)]\) [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklov 57, 64-84 (1976; Zbl 0355.12012) and ibid. 46, 14-35 (1974; Zbl 0345.12009)]. This makes the associated order of \(I\) local. These ingredients are combined with some further ideas in order to prove the main result. Several applications are given. Firstly, it is a direct consequence that all ramification numbers are congruent to \(-1\) modulo \(p^n\) if \(S={\mathcal O}_L\) is Hopf Galois over \(R={\mathcal O}_K\). This is a pleasant generalization of previous results in the case \([L:K]=p^2\) due to \textit{N. P. Byott} [Math. Z. 220, No. 4, 495-522 (1995; Zbl 0841.16021)], and to \textit{C. Greither} [Math. Z. 210, No. 1, 37-68 (1992; Zbl 0737.11038)]. Secondly, the author considers division fields \(K=k_r\) and \(L=k_{m+r}\) associated to a Lubin-Tate group. Then \(L/K\) carries a priori two Hopf Galois structures: the first is the structure of a classical Galois extension, coming simply from …
Mathematics Subject Classification ID: 11S23
Mathematics Subject Classification ID: 11S15
Mathematics Subject Classification ID: 11S31
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 1132894
zbMATH Keywords: associated orders
zbMATH Keywords: Galois modules
zbMATH Keywords: ramification
zbMATH Keywords: Hopf Galois extensions
zbMATH Keywords: \(p\)-adic field
zbMATH Keywords: finite abelian \(p\)-extension
zbMATH Keywords: Kummer theory
reviewed by: Cornelius Greither
MaRDI profile type: Publication
OpenAlex ID: W2312458027
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DOI: 10.5802/JTNB.196
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description: scientific article; zbMATH DE number 1132894