On Galois structure of the integers in cyclic extensions of local number fields
type: Item
instance of: scholarly article
title: On Galois structure of the integers in cyclic extensions of local number fields
zbMATH Open document ID: 1026.11083
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2003-06-30T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/224572
review text: This article presents considerable progress on the following problem: For a Galois extension \(L/K\) of \(p\)-adic fields whose Galois group \(G\) is cyclic of \(p\)-power order \(p^n\), find the decomposition of the \({\mathbb Z}_p[G]\)-module \(O_L\) into indecomposable summands. By E. Noether's result, this module is free iff \(L/K\) is tamely ramified, so the interesting case is when \(L/K\) is totally (hence wildly) ramified. Previous work by Rzedowski-Caldéron, Villa-Salvador, Madan, and by Elder, Madan respectively, considered the cases \(n=2\), respectively, \(n\) general, both times with essential restrictions on the first ramification number \(b_1=b_1(L/K)\). (A little context: If \(b_1\) takes on its maximal value \(pe_0/(p-1)\) or one less, then all idempotents of the algebra \(K[G]\) act on \(O_L\), so this module is free rank one over the maximal order of \(O_K[G]\) in \(K[G]\), which completely solves the problem.) In the present article, \(b_1\) is, loosely speaking, restricted to the upper half of its possible range, and the main theorem 1.1 gives the indecomposable decomposition of \(O_L\) in a formulation which is (quite reasonably, to reduce complexity) recursive in \(n\). It should be pointed out that the removal of summands (``direct difference'') does make sense, given uniqueness of decomposition due to the Krull-Schmidt theorem. The result, and the whole theory, are intrisincally complex and technical. The author does a good job of explaining the involved modules, doing the necessary bookkeeping, and comparing his notation to other work. All the same, the technical parts of the paper are quite challenging, and the reviewer has not gone through the details. There are at least two features of the main result which inspire confidence in spite of the technical complications: firstly, one retrieves, in a nice way, the earlier results for \(n\leq 2\), and secondly the formula of the main result is consistent with passing from \(n\) to \(n-1\): if \…
Mathematics Subject Classification ID: 11R33
zbMATH DE Number: 1943331
author: Gove Griffith Elder
reviewed by: Cornelius Greither
MaRDI profile type: Publication
OpenAlex ID: W2082555140
cites work: Sur les extensions cycliques de degré $p^n$ d'un corps local
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DOI: 10.5802/JTNB.350
description: scientific article; zbMATH DE number 1943331