  We study the question which henselian fields admit definable henselian
valuations (with or without parameters). We show that every field which 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 0-definable (non-trivial)
henselian valuations. We then draw some Galois-theoretic conclusions from our
results.
