数域上的参数扩张

摘要：

给定一个数域$F$，一个有限群$G$和一个不定$T$，{ \it{a $G$-parametric extension over $F$} }是一个有限的伽罗瓦扩张$E/F(T)$，它具有伽罗瓦群$G$和$E/F$正则，在它的专门化中具有$G$的所有$F$的伽罗瓦扩张。我们主要感兴趣的是生产非$G$参数扩展，这涉及到经典的逆伽罗瓦论的问题，如贝克曼-布莱克问题。建立在以前的论文中开发的策略，我们表明，存在至少一个非$G$参数扩展$F$对于一个给定的非平凡有限群$G$和一个给定的数域$F$的唯一必要条件下，$G$作为伽罗瓦群的伽罗瓦扩展$E/F(T)$与$E/F$定期。

英文摘要：

Given a number field $F$, a finite group $G$ and an indeterminate $T$, {\it{a $G$-parametric extension over $F$}} is a finite Galois extension $E/F(T)$ with Galois group $G$ and $E/F$ regular that has all the Galois extensions of $F$ with Galois group $G$ among its specializations. We are mainly interested in producing non-$G$-parametric extensions, which relates to classical questions in inverse Galois theory like the Beckmann-Black problem. Building on a strategy developed in previous papers, we show that there exists at least one non-$G$-parametric extension over $F$ for a given non-trivial finite group $G$ and a given number field $F$ under the sole necessary condition that $G$ occurs as the Galois group of a Galois extension $E/F(T)$ with $E/F$ regular.

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