Stable Rationality and Cyclicity
David J Saltman

TL;DR
This paper explores the relationship between the stable rationality of the center of generic division algebras and their cyclicity, establishing that non-stable rationality implies non-cyclicity in certain cases.
Contribution
It demonstrates a connection between stable rationality of the center and cyclicity of division algebras of prime degree in characteristic zero fields.
Findings
If the center is not stably rational, then the division algebra is not cyclic.
Establishes a link between two major open questions in division algebra theory.
Results apply when the base field contains a primitive p-th root of unity.
Abstract
There are two outstanding questions about division algebras of prime degree . The first is whether they are cyclic, or equivalently crossed products. The second is whether the center, , of the generic division algebra is stably rational over . When is characteristic 0 and contains a primitive root of one, we show that there is a connection between these two questions. Namely, we show that if is not stably rational then is not cyclic.
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Taxonomy
TopicsRings, Modules, and Algebras · Homotopy and Cohomology in Algebraic Topology · Advanced Topics in Algebra
