有理函数域的值群、剩余域和坏处
Value groups, residue fields and bad places of rational function fields
摘要：
我们分类所有可能的扩展的估值从一个基域$K$的有理函数域在一个或多个变量在$K$。我们确定哪些值组和剩余域可以出现，我们展示了如何构建这些值组和剩余域的扩展。特别地，我们给出了几种扩张的构造，它们的相应的值群和剩余域扩张都不是直接生成的。在一个有理函数域$K(x)$在一个变量的情况下，我们认为相对代数闭包的$K$在亨塞尔化的$K(x)$关于给定的扩展，我们表明，这可以是任何可数生成的可分代数扩张的$K$。在“驯服的情况下”，我们展示了如何确定这个相对代数闭包。最后，我们将我们的方法应用到幂级数域和$p$ -adics。
英文摘要：
We classify all possible extensions of a valuation from a ground field $K$ to a rational function field in one or several variables over $K$. We determine which value groups and residue fields can appear, and we show how to construct extensions having these value groups and residue fields. In particular, we give several constructions of extensions whose corresponding value group and residue field extensions are not finitely generated. In the case of a rational function field $K(x)$ in one variable, we consider the relative algebraic closure of $K$ in the henselization of $K(x)$ with respect to the given extension, and we show that this can be any countably generated separable-algebraic extension of $K$. In the "tame case", we show how to determine this relative algebraic closure. Finally, we apply our methods to power series fields and the $p$-adics.
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