The Capitulation Problem in Certain Pure Cubic Fields
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914078257053696 |
|---|---|
| author | Aouissi, Siham Mayer, Daniel C. |
| author_facet | Aouissi, Siham Mayer, Daniel C. |
| contents | Let \(Γ=\mathbb{Q}(\sqrt[3]{n})\) be a pure cubic field with normal closure \(k=\mathbb{Q}(\sqrt[3]{n},ζ)\), where \(n>1\) denotes a cube free integer, and \(ζ\) is a primitive cube root of unity. Suppose \(k\) possesses an elementary bicyclic \(3\)-class group \(\mathrm{Cl}_3(k)\), and the conductor of \(k/\mathbb{Q}(ζ)\) has the shape \(f\in\lbrace pq_1q_2,3pq,9pq\rbrace\) where \(p\equiv 1\,(\mathrm{mod}\,9)\) and \(q,q_1,q_2\equiv 2,5\,(\mathrm{mod}\,9)\) are primes. It is disproved that there are only two possible capitulation types \(\varkappa(k)\), either type \(\mathrm{a}.1\), \((0000)\), or type \(\mathrm{a}.2\), \((1000)\). Evidence is provided, theoretically and experimentally, of two further types, \(\mathrm{b}.10\), \((0320)\), and \(\mathrm{d}.23\), \((1320)\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01361 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Capitulation Problem in Certain Pure Cubic Fields Aouissi, Siham Mayer, Daniel C. Number Theory Group Theory 11R16, 11R20, 11R29, 11R37, 20D15 Let \(Γ=\mathbb{Q}(\sqrt[3]{n})\) be a pure cubic field with normal closure \(k=\mathbb{Q}(\sqrt[3]{n},ζ)\), where \(n>1\) denotes a cube free integer, and \(ζ\) is a primitive cube root of unity. Suppose \(k\) possesses an elementary bicyclic \(3\)-class group \(\mathrm{Cl}_3(k)\), and the conductor of \(k/\mathbb{Q}(ζ)\) has the shape \(f\in\lbrace pq_1q_2,3pq,9pq\rbrace\) where \(p\equiv 1\,(\mathrm{mod}\,9)\) and \(q,q_1,q_2\equiv 2,5\,(\mathrm{mod}\,9)\) are primes. It is disproved that there are only two possible capitulation types \(\varkappa(k)\), either type \(\mathrm{a}.1\), \((0000)\), or type \(\mathrm{a}.2\), \((1000)\). Evidence is provided, theoretically and experimentally, of two further types, \(\mathrm{b}.10\), \((0320)\), and \(\mathrm{d}.23\), \((1320)\). |
| title | The Capitulation Problem in Certain Pure Cubic Fields |
| topic | Number Theory Group Theory 11R16, 11R20, 11R29, 11R37, 20D15 |
| url | https://arxiv.org/abs/2501.01361 |