Associated orders of certain extensions arising from Lubin-Tate formal groups
type: Item
instance of: scholarly article
title: Associated orders of certain extensions arising from Lubin-Tate formal groups
zbMATH Open document ID: 0902.11052
author: Nigel P. Byott
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1998-12-06T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93701
full work available at URL: jtnb9-2.html
review text: In a previous paper [\textit{N. P. Byott}, J. Théor. Nombres Bordx. 9, No. 1, 201-219 (1997; Zbl 0889.11040)], the author proved that in non-Kummer Lubin-Tate extensions \(L/K\), the ring of integers in \(L\) is almost never free over its associated order in \(K[G(L/K)]\). His clever proof did not require an explicit description of the associated order. The present paper can be considered as a complement: here the author considers the minimal non-Kummer example (\(L=\) field of \(\pi^3\)-torsion points, \(K=\) field of \(\pi\)-torsion points), and he gives a complete and utterly explicit calculation of the associated order, under a hypothesis concerning the absolute ramification degree. To do this, he starts from some sums in the group ring \(K[G(L/K)]\) which have also been considered by Taylor; one major point is to find the maximal power of \(\pi_K\) dividing one of these sums in the associated order. This then becomes very technical: among other things, some divisibility properties of binomial coefficients have to be studied. This is a highly specialized paper, but again, a well-written one.
Mathematics Subject Classification ID: 11S23
Mathematics Subject Classification ID: 11S31
Mathematics Subject Classification ID: 11R33
zbMATH DE Number: 1164097
zbMATH Keywords: associated orders
zbMATH Keywords: Galois modules
zbMATH Keywords: ramification
zbMATH Keywords: Lubin-Tate extensions
zbMATH Keywords: divisibility properties of binomial coefficients
reviewed by: Cornelius Greither
MaRDI profile type: Publication
OpenAlex ID: W2320338542
cites work: Some self-dual local rings of integers not free over their associated orders
cites work: Galois structure of ideals in wildly ramified abelian \(p\)-extensions of a \(p\)-adic field, and some applications
cites work: The associated orders of rings of integers in Lubin-Tate division fields over the \(p\)-adic number field
cites work: Q4039755
cites work: Q5094820
cites work: Formal groups and the Galois module structure of local rings of integers.
DOI: 10.5802/JTNB.212
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description: scientific article; zbMATH DE number 1164097