Value groups, residue fields, and bad places of rational function fields
By Franz-Viktor Kuhlmann. 2004
98 citations recorded by the source corpus.
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.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/1a34e1bdb80362c6f5e08ba3113ae146a7ac3896
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- The algebra and model theory of tame valued fields (2013, background): “of [23], there is a valued field (K0, v) of characteristic exponent p with value group Γ and residue field k, and admitting a standard valuation transcendence basis T over its prime field.”