Kummer subfields of tame division algebras
The division algebras \(D\) considered in this paper are finite-dimensional over their center \(F\) and endowed with a (Schilling) valuation whose restriction to \(F\) is Henselian. It is further assumed that \(D=S\otimes T\) for some central division subalgebras \(S\), \(T\), where \(S\) is split by some inertial extension of \(F\) and \(T\) is totally ramified over \(F\) (the cases \(S=F\) or \(T=F\) are allowed) and that the residue characteristic does not divide the degree of \(D\). Under these hypotheses, the authors provide a homological description of the subfields of \(D\) which are Kummer (Galois) extensions of \(F\), generalizing results obtained by \textit{S. A. Amitsur} and the reviewer for certain division algebras of twisted formal power series [Isr. J. Math. 50, 114-144 (1985; Zbl 0564.16015)]. As an application of this description, they show that for any prime \(p\) and any \(n>m\geq 2\) the generic division algebra of degree \(p^n\) and exponent \(p^m\) remains a noncrossed product under any scalar extension of degree prime to \(p\). When \(D\) is a tensor product of two quaternion algebras and the residue field of \(F\) contains a primitive 4-th root of unity, they also provide a necessary and sufficient condition for \(D\) to be cyclic.
scientific article; zbMATH DE number 763206
scientific article