Stable Rationality and Cyclicity

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 $p$. The first is whether they are cyclic, or equivalently crossed products. The second is whether the center, $Z(F,p)$, of the generic division algebra $UD(F,p)$ is stably rational over $F$. When $F$ is characteristic 0 and contains a primitive $p$ root of one, we show that there is a connection between these two questions. Namely, we show that if $Z(F,p)$ is not stably rational then $UD(F,p)$ is not cyclic.