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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.11773 |
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| _version_ | 1866908772716249088 |
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| author | Benjamin, Elliot Chems-Eddin, Mohamed Mahmoud |
| author_facet | Benjamin, Elliot Chems-Eddin, Mohamed Mahmoud |
| contents | In this article we continue the investigation of the length of the narrow $2$-class field tower of real quadratic number fields $\mathrm{k}$ whose discriminants are not a sum of two squares and for which their $2$-class groups are elementary of order $4$. Letting $\mathrm{G}$ equal the Galois group of the second Hilbert narrow $2$-class field over $\mathrm{k}$, and $[\mathrm{G}_i]$ denote the lower central series of $\mathrm{G}$, we give heuristic evidence that the length of the narrow $2$-class field tower of $\mathrm{k}$ is equal to $2$ when $\mathrm{G}/\mathrm{G}_3$ is of type $64.150$ (in the tables of Hall and Senior). We also give the formulation of the relevant unit groups of the narrow Hilbert $2$-class field for these fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_11773 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Narrow 2-Class Field Tower of Some Real Quadratic Number Fields: Lengths Heuristics Follow-Up Benjamin, Elliot Chems-Eddin, Mohamed Mahmoud Number Theory In this article we continue the investigation of the length of the narrow $2$-class field tower of real quadratic number fields $\mathrm{k}$ whose discriminants are not a sum of two squares and for which their $2$-class groups are elementary of order $4$. Letting $\mathrm{G}$ equal the Galois group of the second Hilbert narrow $2$-class field over $\mathrm{k}$, and $[\mathrm{G}_i]$ denote the lower central series of $\mathrm{G}$, we give heuristic evidence that the length of the narrow $2$-class field tower of $\mathrm{k}$ is equal to $2$ when $\mathrm{G}/\mathrm{G}_3$ is of type $64.150$ (in the tables of Hall and Senior). We also give the formulation of the relevant unit groups of the narrow Hilbert $2$-class field for these fields. |
| title | On the Narrow 2-Class Field Tower of Some Real Quadratic Number Fields: Lengths Heuristics Follow-Up |
| topic | Number Theory |
| url | https://arxiv.org/abs/2601.11773 |