关于由关键多项式生成的简单扩张
On simple extensions generated by key polynomials
- University of Delhi（德里大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明关键多项式生成的简单扩张在亨泽尔化不可约时具有单分支性和亏格独立性，推广亨泽尔赋值域结果至一般赋值域，并给出无亏格判据及 Ore 定理的推广。
AI中文摘要：
设 $K^h$ 为赋值域 $(K,v)$ 的亨泽尔化，$w$ 为 $v$ 到 $K[x]$ 的赋值超越扩张，$\phi$ 为 $w$ 的关键多项式。本文证明：若 $\phi$ 在 $K^h$ 上不可约，则由 $w$ 的任意关键多项式的根生成的简单扩张 $L|K$ 是单分支的，且其亏格与关键多项式的选择无关。作为推论，我们将亨泽尔赋值域的若干已知结果推广到任意赋值域。此外，我们利用 Mac Lane-Vaquié 链、抽象关键多项式的完备序列及饱和区分链，给出了 $L|K$ 为单分支且无亏格的若干判据。作为应用，我们推广了 Ore 关于数域存在 $p$-正则生成元的经典结果。
英文摘要：
Let $K^h$ be a Henselization of a valued field $(K,v),$ $w$ a valuation-transcendental extension of $v$ to $K[x],$ and $ϕ$ a key polynomial for $w$. In this paper, we prove that if $ϕ$ is irreducible over $K^h,$ then the simple extension $L|K,$ generated by a root of any key polynomial for $w,$ is unibranched and its defect is independent of the choice of key polynomial. As a consequence, we generalize some well-known results for Henselian valued fields to arbitrary valued fields. Moreover, we provide some criteria for $L|K$ to be unibranched and defectless in terms of Mac Lane-Vaquié chains, complete sequences of abstract key polynomials and saturated distinguished chains. As an application, we generalize a classical result of Ore about the existence of $p$-regular generators for number fields.
