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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2503.16830 |
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| _version_ | 1866917964508299264 |
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| author | Elder, G. Griffith Keating, Kevin |
| author_facet | Elder, G. Griffith Keating, Kevin |
| contents | Let $K=k((t))$ be a local field of characteristic $p>0$, with perfect residue field $k$. Let $\vec{a}=(a_0,a_1,\dots,a_{n-1})\in W_n(K)$ be a Witt vector of length $n$. Artin-Schreier-Witt theory associates to $\vec{a}$ a cyclic extension $L/K$ of degree $p^i$ for some $i\le n$. Assume that the vector $\vec{a}$ is ``reduced'', and that $v_K(a_0)<0$; then $L/K$ is a totally ramified extension of degree $p^n$. In the case where $k$ is finite, Kanesaka-Sekiguchi and Thomas used class field theory to explicitly compute the upper ramification breaks of $L/K$ in terms of the valuations of the components of $\vec{a}$. In this note we use a direct method to show that these formulas remain valid when $k$ is an arbitrary perfect field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16830 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Artin-Schreier-Witt extensions and ramification breaks Elder, G. Griffith Keating, Kevin Number Theory 11S15 Let $K=k((t))$ be a local field of characteristic $p>0$, with perfect residue field $k$. Let $\vec{a}=(a_0,a_1,\dots,a_{n-1})\in W_n(K)$ be a Witt vector of length $n$. Artin-Schreier-Witt theory associates to $\vec{a}$ a cyclic extension $L/K$ of degree $p^i$ for some $i\le n$. Assume that the vector $\vec{a}$ is ``reduced'', and that $v_K(a_0)<0$; then $L/K$ is a totally ramified extension of degree $p^n$. In the case where $k$ is finite, Kanesaka-Sekiguchi and Thomas used class field theory to explicitly compute the upper ramification breaks of $L/K$ in terms of the valuations of the components of $\vec{a}$. In this note we use a direct method to show that these formulas remain valid when $k$ is an arbitrary perfect field. |
| title | Artin-Schreier-Witt extensions and ramification breaks |
| topic | Number Theory 11S15 |
| url | https://arxiv.org/abs/2503.16830 |