The set of division algebras central and finite dimensional over a field F Ε½ . are nicely parameterized by the Brauer group Br F , which is naturally 2 Ε½ β
· . isomorphic to the Galois cohomology group H G , F . Since the latter F sep is an arithmetic invariant, the theory of F's division algebras and Brauer group is a reflection of F's arithmetic complexity. This invariant is so sensitive, however, that a good theory is rarely tractable. This paper treats the strictly henselian case, in characteristic zero, by reducing the problem to linear algebra involving the group of alternate matrices over β«.ήβͺrβ¬ήβ¬ A good theory's first task is to provide a reasonable description of each Brauer class. Then it should be able to find a presentation for a given class's underlying division algebra and to understand the class's splitting behaΒ¨ior with respect to cohomological restriction and corestriction. This amounts to a determination of Br as a functor. The usual reasonable Brauer class description is the presentation of a representative central simple F-algebra. That is, by Wedderburn's theorem, the presentation of a matrix ring over the underlying division algebra.
Suppose F is strictly henselian. For example, F could be the Amitsur field of iterated power series β«ήβ¬ t t ΠΈΠΈΠΈ t .