Kummer subfields of Malcev-Neumann division algebras
Let \(Z^ n\) be a free abelian group with a total ordering. The Malcev- Neumann construction with a suitable co-cycle f can then be used to obtain a skew field \(D_ f\) that is finite-dimensional over its centre \(C_ f\). The authors study the Kummer subfields of \(D_ f\) and use this construction to obtain a division algebra, all of whose maximal subfields are Galois over the centre, with elementary abelian Galois group. These facts are applied to obtain the following improvement on the result of \textit{S. A. Amitsur} [ibid. 12, 408-420 (1972; Zbl 0248.16006)]. Let K be an infinite field of characteristic not dividing n and denote by UD(k,n) the universal division algebra of degree n. Write \(n=m'm''\) where \((m',m'')=1\), m'' is square-free, q is the product of the distinct primes dividing m' and \(q^ 2/n\). If H is a splitting group of UD(k,n), then \(| H| \geq n.m'q^{-2}\). This shows e.g. if n is divisible by the cube of a prime, then UD(k,n) is not a crossed product. Another more precise (but more complicated) lower bound is given for the orders of abelian splitting groups of UD(k,n).
scientific article; zbMATH DE number 3899096
scientific article