  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.
