On the Symbol Length of Fields with finite Square Class Number
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866929333021442048 |
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| author | Hoffmann, Detlev Lorenz, Nico |
| author_facet | Hoffmann, Detlev Lorenz, Nico |
| contents | Let $F$ be a field of characteristic not $2$ with finitely many square classes. Using combinatorial arguments applied to objects related to vector spaces over finite fields, we deduce an upper bound for the number of Pfister forms over $F$. Moreover, we compute upper bounds for the $n$-symbol length $F$ ($n\in\mathbb N$), i.e., the smallest integer $\mathrm{sl}_n(F)\geq 0$ such that to each quadratic form $ϕ\in \mathsf I^n(F)$ there exists some $0\leq k\leq \mathrm{sl}_n(F)$ and Pfister forms $π_1,\ldots, π_k$ such that $φ\equiv π_1+\ldots+π_k\mod \mathsf I^{n+1}(F)$. In particular, we rediscover a bound that can also be deduced from a result by Bruno Kahn that he stated without proof. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_12593 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the Symbol Length of Fields with finite Square Class Number Hoffmann, Detlev Lorenz, Nico Number Theory 11E04, 11E81, 12D15, 12G05, 19D45 Let $F$ be a field of characteristic not $2$ with finitely many square classes. Using combinatorial arguments applied to objects related to vector spaces over finite fields, we deduce an upper bound for the number of Pfister forms over $F$. Moreover, we compute upper bounds for the $n$-symbol length $F$ ($n\in\mathbb N$), i.e., the smallest integer $\mathrm{sl}_n(F)\geq 0$ such that to each quadratic form $ϕ\in \mathsf I^n(F)$ there exists some $0\leq k\leq \mathrm{sl}_n(F)$ and Pfister forms $π_1,\ldots, π_k$ such that $φ\equiv π_1+\ldots+π_k\mod \mathsf I^{n+1}(F)$. In particular, we rediscover a bound that can also be deduced from a result by Bruno Kahn that he stated without proof. |
| title | On the Symbol Length of Fields with finite Square Class Number |
| topic | Number Theory 11E04, 11E81, 12D15, 12G05, 19D45 |
| url | https://arxiv.org/abs/2101.12593 |