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

TL;DR

This paper extends the Hasse invariant concept to the tame Brauer group of higher local fields, enabling new insights into the arithmetic of central simple algebras and their period-index bounds.

Contribution

It generalizes the Hasse invariant to higher local fields and analyzes the tame Brauer group's structure and properties in this context.

Findings

Computed the tame Brauer dimension and period-index bounds.

Determined the cyclic length of henselian-valued fields with finite rank.

Extended local class field theory concepts to higher-dimensional settings.

Abstract

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.