On Crossed Product Algebras over Henselian Valued Fields

Abstract

Let D be a tame central division algebra over a Henselian valued field E, [Formula: see text] be the residue division algebra of D, [Formula: see text] be the residue field of E, and n be a positive integer. We prove that M n ([Formula: see text]) has a strictly maximal subfield which is Galois (resp., abelian) over [Formula: see text] if and only if M n (D) has a strictly maximal subfield K which is Galois (resp., abelian) and tame over E with Γ K ⊆ Γ D , where Γ K and Γ D are the value groups of K and D, respectively. This partially generalizes the result proved by Hanke et al. in 2016 for the case n = 1.