论文标题
田野的一阶理论及其倒数问题
First-order theory of a field and its Inverse Galois Problem
论文作者
论文摘要
令$ g $为有限的组。然后存在一个没有参数的戒指语言的一阶语句$ s（g）$，仅根据$ g $，因此，对于任何字段$ k $，我们都有$ k \型号s（g）$，并且仅当$ k $与Galois Group Isomorphic to $ g $相关时，只有$ k $具有GALOIS扩展。此外，还有一个有效的过程将$ g $的乘法表作为其输入，并产生$ s_g $。因此，鉴于$ k $的字段$ k $，$ k $的倒数galois问题，即确定$ k $是否具有特定galois组的galois扩展名的问题，它可以简化为$ k $的一阶理论。有限的拆分嵌入问题和逆自身形态问题的相似结果。
Let $G$ be a finite group. Then there exists a first-order statement $S(G)$ in the language of rings without parameters and depending only on $G$ such that, for any field $K$, we have that $K\models S(G)$ if and only if $K$ has a Galois extension with the Galois group isomorphic to $G$. Further, there is an effective procedure which takes the table of multiplication of $G$ as its input and produces $S_G$. Therefore, given a field $K$, the Inverse Galois Problem for $K$, that is, the problem of deciding whether $K$ has a Galois extension with a particular Galois group as input, is Turing reducible to the first-order theory of $K$. Similar results hold for the Finite Split Embedding Problem and the Inverse Automorphism Problem.
