  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 over such fields, which are given {\it a
priori} as tensor products of standard cyclic algebras.
  We also compute the tame Brauer dimension (or {\it period-index bound}) and
the cyclic length of a general henselian-valued field of finite rank and finite
residue field.
