Noncrossed product bounds over Henselian fields

Hanke, Timo; Neftin, Danny; Sonn, Jack

doi:10.2140/ant.2014.8.837

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

Noncrossed product bounds over Henselian fields

Timo Hanke, Danny Neftin and Jack Sonn

Vol. 8 (2014), No. 4, 837–855

DOI: 10.2140/ant.2014.8.837

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 $ℚ ((t))$ , 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 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.

Keywords

noncrossed product, division algebra, Henselian fields

Mathematical Subject Classification 2010

Primary: 16S35

Secondary: 11R32, 12F12

Milestones

Received: 1 November 2012

Revised: 28 October 2013

Accepted: 5 December 2013

Published: 10 August 2014

Authors

Timo Hanke
Lehrstuhl D für Mathematik Rheinisch-Wesfälische Technische Hochschule Aachen Templergraben 64 D-52062 Aachen Germany

Danny Neftin
Department of Mathematics University of Michigan 530 Church St. Ann Arbor, MI 48109-1043 United States

Jack Sonn
Department of Mathematics Technion–Israel Institute of Technology Technion City 32000 Haifa Israel

Vol. 8, No. 4, 2014