Value groups, residue fields and bad places of rational function fields

Details Value groups, residue fields and bad places of rational function fields Value groups, residue fields and bad places of rational function fields

2010-03-29

arxiv.org/abs/1003.5685v1

Trans. Amer. Math. Soc. 356 (2004), 4559-4600

Mathematics

Commutative Algebra

41 pages

Scientific paper

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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