2012/10/29 by Franziska Jahnke, Jahnke, Franziska, Jochen Koenigsmann +1 · 1 citation
Mathematics · #03C40 #12E30 #12J10 (Primary) 03C60 #12L12 (Secondary) #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1210.7615
openalex publication_date 2012/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this note we investigate the question whether a henselian valued field carries a non-trivial 0-definable henselian valuation (in the language of rings). It follows from the work of Prestel and Ziegler that there are henselian valued fields which do not admit a 0-definable non-trivial henselian valuation. We give conditions on the residue field which ensure the existence of a parameter-free definiton. In particular, we show that a henselian valued field admits a non-trivial 0-definable valuation when the residue field is separably closed or sufficiently non-henselian, or when the absolute Galois group of the (residue) field is non-universal.