Algebra & Number Theory

Noncrossed product bounds over Henselian fields

Timo Hanke, Danny Neftin, and Jack Sonn

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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 ((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.

Article information

Algebra Number Theory, Volume 8, Number 4 (2014), 837-855.

Received: 1 November 2012
Revised: 28 October 2013
Accepted: 5 December 2013
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16S35: Twisted and skew group rings, crossed products
Secondary: 11R32: Galois theory 12F12: Inverse Galois theory

noncrossed product division algebra Henselian fields


Hanke, Timo; Neftin, Danny; Sonn, Jack. Noncrossed product bounds over Henselian fields. Algebra Number Theory 8 (2014), no. 4, 837--855. doi:10.2140/ant.2014.8.837.

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