Submission date: 29 October 2012.

Abstract:

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.

Mathematics Subject Classification: 03C40, 12E30, 12J10 (Primary) 03C60, 12L12 (Secondary)

Keywords and phrases:

Full text arXiv 1210.7615: pdf, ps.