On the maximal unramified pro-2-extension of $\mathbb{Z}_2$-extension of certain real biquadratic fields
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912037913755648 |
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| author | Chems-Eddin, Mohamed Mahmoud |
| author_facet | Chems-Eddin, Mohamed Mahmoud |
| contents | For any positive integer $n$, we show that there exists a real number field $k$ (resp. $k'$) of degree $2^n$ whose $2$-class group is isomorphic $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ such that the Galois group of the maximal unramified extension of $k$ (resp. $k'$) over $k$ (resp. $k'$) is abelian (resp. non abelian, more precisely isomorphic to $Q_8$ or $D_8$, the quaternion and the dihedral group of order $8$ respectively). In fact, we construct the first examples in literature of families of real biquadratic fields whose unramified abelian Iwasawa module is isomorphic to $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, and so that satisfying the Greenberg conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13574 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the maximal unramified pro-2-extension of $\mathbb{Z}_2$-extension of certain real biquadratic fields Chems-Eddin, Mohamed Mahmoud Number Theory For any positive integer $n$, we show that there exists a real number field $k$ (resp. $k'$) of degree $2^n$ whose $2$-class group is isomorphic $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ such that the Galois group of the maximal unramified extension of $k$ (resp. $k'$) over $k$ (resp. $k'$) is abelian (resp. non abelian, more precisely isomorphic to $Q_8$ or $D_8$, the quaternion and the dihedral group of order $8$ respectively). In fact, we construct the first examples in literature of families of real biquadratic fields whose unramified abelian Iwasawa module is isomorphic to $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, and so that satisfying the Greenberg conjecture. |
| title | On the maximal unramified pro-2-extension of $\mathbb{Z}_2$-extension of certain real biquadratic fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2409.13574 |