Depth of extensions of valuations
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866929738563452928 |
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| author | Novacoski, Josnei Nart, Enric |
| author_facet | Novacoski, Josnei Nart, Enric |
| contents | In this paper we develop the theory of the depth of a simple algebraic extension of valued fields $(L/K,v)$. This is defined as the minimal number of augmentations appearing in some Mac Lane-Vaquié chain for the valuation on $K[x]$ determined by the choice of some generator of the extension. In the defectless and unibranched case, this concept leads to a generalization of a classical result of Ore about the existence of $p$-regular generators for number fields. Also, we find what valuation-theoretic conditions characterize the extensions having depth one. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_00850 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Depth of extensions of valuations Novacoski, Josnei Nart, Enric Commutative Algebra In this paper we develop the theory of the depth of a simple algebraic extension of valued fields $(L/K,v)$. This is defined as the minimal number of augmentations appearing in some Mac Lane-Vaquié chain for the valuation on $K[x]$ determined by the choice of some generator of the extension. In the defectless and unibranched case, this concept leads to a generalization of a classical result of Ore about the existence of $p$-regular generators for number fields. Also, we find what valuation-theoretic conditions characterize the extensions having depth one. |
| title | Depth of extensions of valuations |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2503.00850 |