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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.15947 |
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| _version_ | 1866912109994967040 |
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| author | Wang, Shanwen Yuan, Yijun |
| author_facet | Wang, Shanwen Yuan, Yijun |
| contents | Let $p\geq 3$ be a prime. The hyper-algebraic elements in the $p$-adic Mal'cev-Neumann field $\mathbb{L}_p$ form an algebraically closed subfield $\mathbb{L}_p^{\operatorname{ha}}$. In this article, we clarify the relations among the fields $\mathbb{L}_p^{\operatorname{ha}}$, $\overline{\mathbb{Q}}_p$ and $\mathbb{C}_p$. We introduce two arithmetic invariants (hyper-tame index and hyper-inertia index) of hyper-algebraic elements and study the relation between these invariants and classical arithmetic invariants of $p$-adic algebraic numbers. Finally, we give a criterion for hyper-algebraic elements to be tamely ramified over $\mathbb{Q}_p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15947 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hyper-algebraic invariants of $p$-adic algebraic numbers Wang, Shanwen Yuan, Yijun Number Theory 11S15, 11D88, 41A58, 16W60 Let $p\geq 3$ be a prime. The hyper-algebraic elements in the $p$-adic Mal'cev-Neumann field $\mathbb{L}_p$ form an algebraically closed subfield $\mathbb{L}_p^{\operatorname{ha}}$. In this article, we clarify the relations among the fields $\mathbb{L}_p^{\operatorname{ha}}$, $\overline{\mathbb{Q}}_p$ and $\mathbb{C}_p$. We introduce two arithmetic invariants (hyper-tame index and hyper-inertia index) of hyper-algebraic elements and study the relation between these invariants and classical arithmetic invariants of $p$-adic algebraic numbers. Finally, we give a criterion for hyper-algebraic elements to be tamely ramified over $\mathbb{Q}_p$. |
| title | Hyper-algebraic invariants of $p$-adic algebraic numbers |
| topic | Number Theory 11S15, 11D88, 41A58, 16W60 |
| url | https://arxiv.org/abs/2402.15947 |