Valuated division algebras and crossed products
The author's aim is to extend Amitsur's result on non-crossed products [\textit{S. A. Amitsur}, Isr. J. Math. 12, 408-420 (1972; Zbl 0248.16006)]. Let \(U=UD(K,n,t)\) be the generic division algebra of t \((\geq 2)\) \(n\times n\) matrices over a field K. Suppose that K has a rank 1 valuation, with residue class field \(\tilde K.\) He proves that if U is a crossed product, then so is \(\tilde U= UD(\tilde K,n,t)\); moreover if char\((\tilde K)\) does not divide n and U is a crossed product with a group G, then \(\tilde U\) is a crossed product with G. Combined with Amitsur's theorem this shows that if \(UD(Q,n,t)\) is a crossed product, then any division algebra D of dimension \(n^ 2\) over its centre is a crossed product (with the same group in case char\((D) \nmid n)\). The proof is by an analysis of crossed products using a result of Procesi on generic division algebras.
scientific article; zbMATH DE number 3995908