Reduced norm map of division algebras over complete discrete valuation fields of certain type

We study a ramification theory for a division algebra D of the following type: The center of D is a complete discrete valuation field K with the imperfect residue field F of certain type, and the residue algebra of D is commutative and purely inseparable over F. Using Swan conductors of the corresponding element of Br(K), we define Herbrand's ψ-function of D/K, and it describes the action of the reduced norm map on the filtration of D*. We also gives a relation among the Swan conductors and the 'ramification number' of D, which is defined by the commutator group of D*.