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| Format: | Preprint |
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2015
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1503.05803 |
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| _version_ | 1866909206255239168 |
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| author | Anscombe, Sylvy |
| author_facet | Anscombe, Sylvy |
| contents | In this note we study one-dimensional definable sets in power series fields with perfect residue fields. Using the description of automorphisms given by Schilling, in \cite{S44}, we show that such sets are unions of existentially definable in the language of rings, allowing parameters. We deduce that if $F$ is a perfect field of positive characteristic $p$, and $X$ is a subset of the $t$-adically valued $F((t))$ that is definable in the language of valued fields with parameters from $F$, then the subfield $(X)$ generated by $X$ is either contained in $F$ or equal to $F((t^{p^n}))$, for some $n\geq0$. The proof uses our earlier work on existentially definable subsets of henselian and large fields, of which power series fields are examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1503_05803 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | One-dimensional F-definable sets in F((t)) Anscombe, Sylvy Logic In this note we study one-dimensional definable sets in power series fields with perfect residue fields. Using the description of automorphisms given by Schilling, in \cite{S44}, we show that such sets are unions of existentially definable in the language of rings, allowing parameters. We deduce that if $F$ is a perfect field of positive characteristic $p$, and $X$ is a subset of the $t$-adically valued $F((t))$ that is definable in the language of valued fields with parameters from $F$, then the subfield $(X)$ generated by $X$ is either contained in $F$ or equal to $F((t^{p^n}))$, for some $n\geq0$. The proof uses our earlier work on existentially definable subsets of henselian and large fields, of which power series fields are examples. |
| title | One-dimensional F-definable sets in F((t)) |
| topic | Logic |
| url | https://arxiv.org/abs/1503.05803 |