Valuation rings in simple algebraic extensions of valued fields
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866910871705354240 |
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| author | Novacoski, Josnei Spivakovsky, Mark |
| author_facet | Novacoski, Josnei Spivakovsky, Mark |
| contents | Consider a simple algebraic valued field extension $(L/K,v)$ and denote by $\mathcal O_L$ and $\mathcal O_K$ the corresponding valuation rings. The main goal of this paper is to present, under certain assumptions, a description of $\mathcal O_L$ in terms of generators and relations over $\mathcal O_K$. The main tool used here are complete sequences of key polynomials. It is known that if the ramification index of $(L/K,v)$ is one, then every complete set gives rise to a set of generators of $\mathcal O_L$ over $\mathcal O_K$. We show that we can find a sequence of key polynomials for $(L/K,v)$ which satisfies good properties (called neat). Then we present explicit ``neat" relations that generate all the relations between the corresponding generators of $\mathcal O_L$ over $\mathcal O_K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09096 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Valuation rings in simple algebraic extensions of valued fields Novacoski, Josnei Spivakovsky, Mark Commutative Algebra Consider a simple algebraic valued field extension $(L/K,v)$ and denote by $\mathcal O_L$ and $\mathcal O_K$ the corresponding valuation rings. The main goal of this paper is to present, under certain assumptions, a description of $\mathcal O_L$ in terms of generators and relations over $\mathcal O_K$. The main tool used here are complete sequences of key polynomials. It is known that if the ramification index of $(L/K,v)$ is one, then every complete set gives rise to a set of generators of $\mathcal O_L$ over $\mathcal O_K$. We show that we can find a sequence of key polynomials for $(L/K,v)$ which satisfies good properties (called neat). Then we present explicit ``neat" relations that generate all the relations between the corresponding generators of $\mathcal O_L$ over $\mathcal O_K$. |
| title | Valuation rings in simple algebraic extensions of valued fields |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2503.09096 |