The finitude of tamely ramified pro-$p$ extensions of number fields with cyclic $p$-class groups
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913233664737280 |
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| author | Lee, Yoonjin Lim, Donghyeok |
| author_facet | Lee, Yoonjin Lim, Donghyeok |
| contents | Let $p$ be an odd prime and $F$ be a number field whose $p$-class group is cyclic. Let $F_{\{\mathfrak{q}\}}$ be the maximal pro-$p$ extension of $F$ which is unramified outside a single non-$p$-adic prime ideal $\mathfrak{q}$ of $F$. In this work, we study the finitude of the Galois group $G_{\{\mathfrak{q}\}}(F)$ of $F_{\{\mathfrak{q}\}}$ over $F$. We prove that $G_{\{\mathfrak{q}\}}(F)$ is finite for the majority of $\mathfrak{q}$'s such that the generator rank of $G_{\{\mathfrak{q}\}}(F)$ is two, provided that for $p = 3$, $F$ is not a complex quartic field containing the primitive third roots of unity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08512 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The finitude of tamely ramified pro-$p$ extensions of number fields with cyclic $p$-class groups Lee, Yoonjin Lim, Donghyeok Number Theory 11R32, 11R37 Let $p$ be an odd prime and $F$ be a number field whose $p$-class group is cyclic. Let $F_{\{\mathfrak{q}\}}$ be the maximal pro-$p$ extension of $F$ which is unramified outside a single non-$p$-adic prime ideal $\mathfrak{q}$ of $F$. In this work, we study the finitude of the Galois group $G_{\{\mathfrak{q}\}}(F)$ of $F_{\{\mathfrak{q}\}}$ over $F$. We prove that $G_{\{\mathfrak{q}\}}(F)$ is finite for the majority of $\mathfrak{q}$'s such that the generator rank of $G_{\{\mathfrak{q}\}}(F)$ is two, provided that for $p = 3$, $F$ is not a complex quartic field containing the primitive third roots of unity. |
| title | The finitude of tamely ramified pro-$p$ extensions of number fields with cyclic $p$-class groups |
| topic | Number Theory 11R32, 11R37 |
| url | https://arxiv.org/abs/2402.08512 |