On the Existence of the Maximal Unramified Pro-$2$-Extension over the Cyclotomic $\mathbb{Z}_2$-Extension with Prescribed Metacyclic Galois Group
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| Format: | Preprint |
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2025
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| author | Chems-Eddin, Mohamed Mahmoud Mamry, Hamza El |
| author_facet | Chems-Eddin, Mohamed Mahmoud Mamry, Hamza El |
| contents | For an integer $m\geq 2$, we aim to investigate the realizability of types of metacyclic-nonmodular groups, whose abelianization is $\mathbb{Z}/2 \mathbb{Z}\times\mathbb{Z}/2^m \mathbb{Z}$, as the Galois group of the maximal unramified $2$-extension (resp. pro-$2$-extension) over certain number fields of $2$-power degree (resp. cyclotomic $\mathbb Z_2$-extensions). Furthermore, we present some new techniques for studying Greenberg's conjecture for some number fields.
In particular, the reader can find results concerning the real quadratic fields $F=\mathbb{Q}(\sqrt{ηq rs})$, the real biquadratic fields $K=\mathbb{Q}(\sqrt{ηq},\sqrt{rs})$, with $η\in\{1,2\}$, and the Fröhlich multiquadratic fields of the form $\mathbb{F}=\mathbb{Q}(\sqrt{q }, \sqrt {r}, \sqrt{s})$, where $q$, $r$ and $s$ are odd prime numbers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_18402 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Existence of the Maximal Unramified Pro-$2$-Extension over the Cyclotomic $\mathbb{Z}_2$-Extension with Prescribed Metacyclic Galois Group Chems-Eddin, Mohamed Mahmoud Mamry, Hamza El Number Theory For an integer $m\geq 2$, we aim to investigate the realizability of types of metacyclic-nonmodular groups, whose abelianization is $\mathbb{Z}/2 \mathbb{Z}\times\mathbb{Z}/2^m \mathbb{Z}$, as the Galois group of the maximal unramified $2$-extension (resp. pro-$2$-extension) over certain number fields of $2$-power degree (resp. cyclotomic $\mathbb Z_2$-extensions). Furthermore, we present some new techniques for studying Greenberg's conjecture for some number fields. In particular, the reader can find results concerning the real quadratic fields $F=\mathbb{Q}(\sqrt{ηq rs})$, the real biquadratic fields $K=\mathbb{Q}(\sqrt{ηq},\sqrt{rs})$, with $η\in\{1,2\}$, and the Fröhlich multiquadratic fields of the form $\mathbb{F}=\mathbb{Q}(\sqrt{q }, \sqrt {r}, \sqrt{s})$, where $q$, $r$ and $s$ are odd prime numbers. |
| title | On the Existence of the Maximal Unramified Pro-$2$-Extension over the Cyclotomic $\mathbb{Z}_2$-Extension with Prescribed Metacyclic Galois Group |
| topic | Number Theory |
| url | https://arxiv.org/abs/2508.18402 |