Additive structure of non-monogenic simplest cubic fields
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866909534855888896 |
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| author | Gil-Muñoz, Daniel Tinková, Magdaléna |
| author_facet | Gil-Muñoz, Daniel Tinková, Magdaléna |
| contents | We consider Shanks' simplest cubic fields $K$ for which the index $[\mathcal{O}_K:\mathbb{Z}[ρ]]$ of a root $ρ$ of the defining parametric polynomial is $3$. For them, we study the additive indecomposables of $K$ and provide a complete list of them. Moreover, we use the knowledge of the indecomposables to prove some interesting consequences on the arithmetic of $K$. Mainly, we obtain good bounds on the ranks of universal quadratic forms over $K$ and prove that the Pythagoras number of $\mathcal{O}_K$ is $6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_00364 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Additive structure of non-monogenic simplest cubic fields Gil-Muñoz, Daniel Tinková, Magdaléna Number Theory 11R16, 11R80, 11R04 We consider Shanks' simplest cubic fields $K$ for which the index $[\mathcal{O}_K:\mathbb{Z}[ρ]]$ of a root $ρ$ of the defining parametric polynomial is $3$. For them, we study the additive indecomposables of $K$ and provide a complete list of them. Moreover, we use the knowledge of the indecomposables to prove some interesting consequences on the arithmetic of $K$. Mainly, we obtain good bounds on the ranks of universal quadratic forms over $K$ and prove that the Pythagoras number of $\mathcal{O}_K$ is $6$. |
| title | Additive structure of non-monogenic simplest cubic fields |
| topic | Number Theory 11R16, 11R80, 11R04 |
| url | https://arxiv.org/abs/2212.00364 |