Defining coarsenings of valuations - Jahnke, F; Koenigsmann, J (Cambridge University Press, 2017)
We study the question of which Henselian fields admit definable Henselian valuations (with or without parameters). We show that every field that admits a Henselian valuation with non-divisible value group admits a parameter-definable (non-trivial) Henselian valuation. In equicharacteristic 0, we give a complete characterization of Henselian fields admitting a parameter-definable (non-trivial) Henselian valuation. We also obtain partial characterization results of fields admitting-definable (non-trivial) Henselian valuations. We then draw some Galois-theoretic conclusions from our results.