A Choice-free look at algebraic extensions of valuations
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914147237625856 |
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| author | Aïd, Cédric |
| author_facet | Aïd, Cédric |
| contents | In this paper, we study extensions of valuations over algebraic field extensions without the use of the Axiom of Choice. We show a bijection between the extensions of a valuation and the maximal ideals of the relative integral closure of its valuation ring. In the case of a finite extension, we show that these maximal ideals exist. We conclude with an elementary proof of the fundamental inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_07177 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Choice-free look at algebraic extensions of valuations Aïd, Cédric Commutative Algebra 12J20 (Primary) 13F30, 12F05 (Secondary) In this paper, we study extensions of valuations over algebraic field extensions without the use of the Axiom of Choice. We show a bijection between the extensions of a valuation and the maximal ideals of the relative integral closure of its valuation ring. In the case of a finite extension, we show that these maximal ideals exist. We conclude with an elementary proof of the fundamental inequality. |
| title | A Choice-free look at algebraic extensions of valuations |
| topic | Commutative Algebra 12J20 (Primary) 13F30, 12F05 (Secondary) |
| url | https://arxiv.org/abs/2511.07177 |