Hasse invariant for the tame Brauer group of a higher local field

We generalize the Hasse invariant of local class field theory to the tame Brauer group of a higher dimensional local field, and use it to study the arithmetic of central simple algebras, which are given a priori as tensor products of standard cyclic algebras. We also compute the tame Brauer dimension (or period-index bound) and the cyclic length of a general henselian-valued field of finite rank and finite residue field.