Noncrossed product bounds over Henselian fields

Abstract

The existence of noncrossed product division algebras (finite-dimensional central division algebras with no maximal subfield that is Galois over the center) was for a time the biggest open problem in the theory of division algebras, until it was settled by Amitsur.

Motivated by Brussel’s discovery of noncrossed products over $\mathbb{Q}\left(\left(t\right)\right)$, we describe the “location” of noncrossed products in the Brauer group of general Henselian valued fields with arbitrary value group and global residue field. We show that within the fibers defined canonically by Witt’s decomposition of the Brauer group of such fields, crossed products and noncrossed products are, roughly speaking, separated by an index bound. This generalizes a result of Hanke and Sonn for rank-$1$ valued Henselian fields.

Furthermore, we show that the new index bounds are of different nature from the rank-$1$ case. In particular, all fibers not covered by the rank-1 case contain noncrossed products, unless the residue characteristic interferes.