Continuation of hereditary orders in local central simple algebras
type: Item
instance of: scholarly article
title: Continuation of hereditary orders in local central simple algebras
zbMATH Open document ID: 0932.16010
author: Martin Grabitz
published in: Journal of Number Theory
publication date: 1999-12-15T00:00:00Z
review text: Let \(A\) be a central simple algebra over a non-Archimedian local field \(K\) with ring of integers \(R\). For a field extension \(L|K\) in \(A\) consider a hereditary order \(\Lambda\) in the centralizer \(B\subset A\) of \(L\). Let \(N_\Lambda\) denote the normalizer of \(\Lambda\) in \(B^\times\). The author proves that \(\Lambda\) admits an extension to \(A\), that is, a hereditary \(R\)-order \(\Gamma\) in \(A\) with \(\Gamma\cap B=\Lambda\) and \(N_\Lambda\subset N_\Gamma\). An equivalent formulation is provided in terms of pseudo-valuations \(A\to\mathbb{Z}\cup\{\infty\}\) in the sense of \textit{H. Benz} [Über eine Bewertungstheorie der Algebren und ihre Bedeutung für die Arithmetik, Berlin (1961; Zbl 0205.06004)]. The proof starts with the fact that a hereditary order \(\Gamma\) can be naturally represented as an intersection of principal orders \(\Gamma_i\), i.e., hereditary orders with principal Jacobson radical. Therefore, the author considers pairs \((\Gamma,\Gamma_i)\). Then he shows that extension of pairs is unique. A related result of \textit{H. Benz} [J. Reine Angew. Math. 225, 30-75 (1967; Zbl 0155.09802)] states that principal orders in a maximal commutative subfield of \(A\) admit a unique extension to a principal order in \(A\). [More general, see also \textit{H. Benz} and \textit{H. Zassenhaus}, J. Number Theory 20, 282-298 (1985; Zbl 0593.16005)].
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 11S45
zbMATH DE Number: 1309263
zbMATH Keywords: central simple algebras
zbMATH Keywords: non-Archimedian local fields
zbMATH Keywords: pseudo-valuations
zbMATH Keywords: principal orders
zbMATH Keywords: hereditary orders
zbMATH Keywords: extensions of pairs
reviewed by: Wolfgang Rump
MaRDI profile type: Publication
full work available at URL: jnth.1999.2374
OpenAlex ID: W2074592168
cites work: Q5604581
cites work: Untersuchungen zur Arithmetik in lokalen einfachen Algebren, insbesondere über maximalen Teilkörpern. I.
cites work: Hereditary orders and embeddings of local fields in simple algebras
cites work: Non-Abelian Congruence Gauss Sums and p -Adic Simple Algebras
cites work: Q4317818
cites work: The admissible dual of ${\rm SL}(N)$. I
cites work: Schémas en groupes et immeubles des groupes classiques sur un corps local
cites work: Q3922829
cites work: Principal Orders and Embedding of Local Fields in Algebras
cites work: Q3996577
cites work: Q4061092
DOI: 10.1006/JNTH.1999.2374
description: scientific article; zbMATH DE number 1309263