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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2308.04099 |
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| _version_ | 1866911090919604224 |
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| author | Lim, Meng Fai |
| author_facet | Lim, Meng Fai |
| contents | Let $k$ be a given positive odd integer and $p$ an odd prime. In this paper, we shall give a sufficient condition when a prime $p$ divides the order of the groups $K_{2k}(\mathbb{Z}[ζ_m+ζ_m^{-1}])$ and $K_{2k}(\mathbb{Z}[ζ_m])$, where $ζ_m$ is a primitive $m$th root of unity. When $F$ is a $p$-extension contained in $\mathbb{Q}(ζ_l)$ for some prime $l$, we also establish a necessary and sufficient condition for the order of $K_{2(p-2)}(\mathcal{O}_F)$ to be divisible by $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_04099 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the $p$-divisibility of even $K$-groups of the ring of integers of a cyclotomic field Lim, Meng Fai Number Theory Let $k$ be a given positive odd integer and $p$ an odd prime. In this paper, we shall give a sufficient condition when a prime $p$ divides the order of the groups $K_{2k}(\mathbb{Z}[ζ_m+ζ_m^{-1}])$ and $K_{2k}(\mathbb{Z}[ζ_m])$, where $ζ_m$ is a primitive $m$th root of unity. When $F$ is a $p$-extension contained in $\mathbb{Q}(ζ_l)$ for some prime $l$, we also establish a necessary and sufficient condition for the order of $K_{2(p-2)}(\mathcal{O}_F)$ to be divisible by $p$. |
| title | On the $p$-divisibility of even $K$-groups of the ring of integers of a cyclotomic field |
| topic | Number Theory |
| url | https://arxiv.org/abs/2308.04099 |