Continuation of hereditary orders in local central simple algebras
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)].
scientific article; zbMATH DE number 1309263
scientific article