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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2411.07880 |
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| _version_ | 1866909386348167168 |
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| author | Dhar, Shreya Newman, River Plumpton, Grayson Wang, Chenglu |
| author_facet | Dhar, Shreya Newman, River Plumpton, Grayson Wang, Chenglu |
| contents | Let $p$ be a prime and let $\mathbb{Q}_p$ be the field of $p$-adic numbers. It is known that the finite extensions of $\mathbb{Q}_p$ of a given degree are finite up to isomorphism. Given a cubic field extension $L$ of $\mathbb{Q}_p$ generated by the root of an irreducible polynomial $h$, we present a practical (closed-form) method to determine the isomorphism class in which $L$ lives, based on the coefficients of $h$. We discuss the subtleties of the wildly ramified case, when the degree of the extension coincides with $p$, the characteristic of the residue field. We also present a method for tamely ramified extensions of arbitrary prime degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07880 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Classifying Extensions of $p$-adic Fields Dhar, Shreya Newman, River Plumpton, Grayson Wang, Chenglu Number Theory 11S15 (Primary), 11S05, 11S20 (Secondary) Let $p$ be a prime and let $\mathbb{Q}_p$ be the field of $p$-adic numbers. It is known that the finite extensions of $\mathbb{Q}_p$ of a given degree are finite up to isomorphism. Given a cubic field extension $L$ of $\mathbb{Q}_p$ generated by the root of an irreducible polynomial $h$, we present a practical (closed-form) method to determine the isomorphism class in which $L$ lives, based on the coefficients of $h$. We discuss the subtleties of the wildly ramified case, when the degree of the extension coincides with $p$, the characteristic of the residue field. We also present a method for tamely ramified extensions of arbitrary prime degree. |
| title | On Classifying Extensions of $p$-adic Fields |
| topic | Number Theory 11S15 (Primary), 11S05, 11S20 (Secondary) |
| url | https://arxiv.org/abs/2411.07880 |