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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.07240 |
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| _version_ | 1866910599766605824 |
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| author | Saltman, David J |
| author_facet | Saltman, David J |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07240 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stable Rationality and Cyclicity Saltman, David J Rings and Algebras Algebraic Geometry 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. |
| title | Stable Rationality and Cyclicity |
| topic | Rings and Algebras Algebraic Geometry |
| url | https://arxiv.org/abs/2409.07240 |