On defectless unibranched simple extensions, complete distinguished chains and certain stability results
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913811757268992 |
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| author | Dutta, Arpan Ghosh, Rumi |
| author_facet | Dutta, Arpan Ghosh, Rumi |
| contents | Let $(K,v)$ be a valued field. Take an extension of $v$ to a fixed algebraic closure $L$ of $K$. In this paper we show that an element $a\in L$ admits a complete distinguished chain over $K$ if and only if the extension $(K(a)|K,v)$ is defectless and unibranched. This characterization generalizes the known result in the henselian case. In particular, our result shows that if $a$ admits a complete distinguished chain over $K$, then it also admits one over the henselization; however, the converse may not be true. The main tool employed in our analysis is the stability of the $j$-invariant associated to a valuation transcendental extension under passage to the henselization.
We also explore the stability of defectless simple extensions in the following sense: let $(K(X)|K,w)$ be a valuation transcendental extension with a pair of definition $(b,γ)$. Assume that either $(K(b)|K,v)$ is a defectless extension, or that $f(X)$ is a key polynomial for $w$ over $K$, where $f(X)$ is the minimal polynomial of $b$ over $K$. We show that then the extension $(K(b,X)|K(X),w)$ is defectless. In particular, the extension $(K(b,X)|K(X),w)$ is always defectless whenever $(b,γ)$ is a minimal pair of definition for $w$ over $K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_07830 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On defectless unibranched simple extensions, complete distinguished chains and certain stability results Dutta, Arpan Ghosh, Rumi Commutative Algebra Algebraic Geometry 12J20, 13A18, 12J25 Let $(K,v)$ be a valued field. Take an extension of $v$ to a fixed algebraic closure $L$ of $K$. In this paper we show that an element $a\in L$ admits a complete distinguished chain over $K$ if and only if the extension $(K(a)|K,v)$ is defectless and unibranched. This characterization generalizes the known result in the henselian case. In particular, our result shows that if $a$ admits a complete distinguished chain over $K$, then it also admits one over the henselization; however, the converse may not be true. The main tool employed in our analysis is the stability of the $j$-invariant associated to a valuation transcendental extension under passage to the henselization. We also explore the stability of defectless simple extensions in the following sense: let $(K(X)|K,w)$ be a valuation transcendental extension with a pair of definition $(b,γ)$. Assume that either $(K(b)|K,v)$ is a defectless extension, or that $f(X)$ is a key polynomial for $w$ over $K$, where $f(X)$ is the minimal polynomial of $b$ over $K$. We show that then the extension $(K(b,X)|K(X),w)$ is defectless. In particular, the extension $(K(b,X)|K(X),w)$ is always defectless whenever $(b,γ)$ is a minimal pair of definition for $w$ over $K$. |
| title | On defectless unibranched simple extensions, complete distinguished chains and certain stability results |
| topic | Commutative Algebra Algebraic Geometry 12J20, 13A18, 12J25 |
| url | https://arxiv.org/abs/2503.07830 |