Noncrossed product bounds over Henselian fields

TL;DR

This paper investigates the distribution of noncrossed product division algebras over Henselian fields, establishing bounds and conditions for their existence across various valuations and residue fields.

Contribution

It generalizes previous results by describing the location of noncrossed products in the Brauer group of Henselian fields with arbitrary value groups and residue fields.

Findings

Noncrossed products are separated from crossed products by an index bound within the Brauer group fibers.

All fibers outside the rank 1 case contain noncrossed products under certain conditions.

Index bounds in higher rank cases are independent of roots of unity, differing from rank 1 scenarios.

Abstract

The existence of finite dimensional central division algebras with no maximal subfield that is Galois over the center (called noncrossed products), was for a time the biggest open problem in the theory of division algebras, before it was settled by Amitsur. Motivated by Brussel's discovery of noncrossed products over Q((t)), 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 the first and third author for rank 1 valued Henselian fields. Furthermore, we prove that all fibers which are not covered by the rank 1 case, and where the characteristic of the residue field does not interfere, contain noncrossed products. We show by example that, unlike in the rank 1 case, the value of the index bound does not depend on the number of roots of unity that are present. Thus, the index bounds are in general of a different nature than in the rank 1 case.