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We generalize the Hasse invariant of local class field theory to the tameBrauer group of a higher dimensional local field, and use it to study thearithmetic of central simple algebras over such fields, which are given {\it apriori} as tensor products of standard cyclic algebras.  We also compute the tame Brauer dimension (or {\it period-index bound}) andthe cyclic length of a general henselian-valued field of finite rank and finiteresidue field.

Hasse Invariant for the Tame Brauer Group of a Higher Local Field