{"title":"Stable Rationality and Cyclicity","authors":"David J Saltman","doi":"arxiv-2409.07240","DOIUrl":null,"url":null,"abstract":"There are two outstanding questions about division algebras of prime degree\n$p$. The first is whether they are cyclic, or equivalently crossed products.\nThe second is whether the center, $Z(F,p)$, of the generic division algebra\n$UD(F,p)$ is stably rational over $F$. When $F$ is characteristic 0 and\ncontains a primitive $p$ root of one, we show that there is a connection\nbetween these two questions. Namely, we show that if $Z(F,p)$ is not stably\nrational then $UD(F,p)$ is not cyclic.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"33 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.07240","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.