一个共变式与七次 Hermite--Joubert 问题
A septic covariant and the Hermite--Joubert problem in degree seven
- Illinois State University（伊利诺伊州立大学）
- The University of Western Ontario（韦仕敦大学（西安大略大学））
- Tempered AI
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明了特征零下 Hermite--Joubert 问题的七次情形，通过显式共变式构造生成元，并宣布了任意域上的一般定理。
AI中文摘要：
我们证明了特征零情形下 Hermite--Joubert 问题的七次情形：若 $F$ 是特征零的域且 $E/F$ 是七次域扩张，则 $E$ 由元素 $a$ 生成，满足 $\text{tr}_{E/F}(a)=\text{tr}_{E/F}(a^{3})=0$，即其极小多项式形如 $\lambda^{7}+c_{2}\lambda^{5}+c_{4}\lambda^{3}+c_{5}\lambda^{2}+c_{6}\lambda+c_{7}$，其中 $c_i$ 属于 $F$。该结果由显式公式给出：二元七次式的一个系数次数为七、阶数为五的共变式，在生成元 $\theta$ 处取值并除以其极小多项式在 $\theta$ 处的导数。我们还宣布了一般定理，该定理将在准备中的姊妹论文中证明：在每个无限域上，在任意特征下，每个七次 étale 代数都包含一个本原元素 $a$，满足 $c_{1}(a)=c_{3}(a)=0$。此外，任意域的每个七次域扩张都有一个生成元 $a$，满足 $c_{1}(a)=c_{3}(a)=0$。
英文摘要：
We prove the degree-seven case of the Hermite--Joubert problem in characteristic zero: if $F$ is a field of characteristic zero and $E/F$ is a field extension of degree seven, then $E$ is generated by an element $a$ with $\text{tr}_{E/F}(a)=\text{tr}_{E/F}(a^{3})=0$, that is, with minimal polynomial of the form $λ^{7}+c_{2}λ^{5}+c_{4}λ^{3}+c_{5}λ^{2}+c_{6}λ+c_{7}$, where the $c_i$'s belong to $F$. This is given by an explicit formula: a covariant of the binary septic of coefficient degree seven and order five, evaluated at a generator $θ$ and divided by the derivative of its minimal polynomial evaluated at $θ$. We also announce the general theorem, which will be proved in a companion paper in preparation: over every infinite field, in every characteristic, every étale algebra of degree seven contains a primitive element $a$ with $c_{1}(a)=c_{3}(a)=0$. Moreover, every field extension of degree seven of an arbitrary field has a generator $a$ with $c_{1}(a)=c_{3}(a)=0$.
