Valuation theory on finite-dimensional division algebras
by
A. R. Wadsworth
University of California, San Diego

Valuation theory on division algebras has a different flavor from valuation theory on fields, mostly because of the paucity of valuations in the noncommutative setting. If D is a division algebra finite dimensional over its center F, then most valuations on v will not extend to D, but if a given v does extend to D, the extension is unique, and the presence of the valuation on D can lead to significant information about the structure of D that is otherwise scarecely accessible. This talk will be a survey of some of the ways that valuation theory has been used in studying finite dimensional division algebras, including constructions of noncrossed products, indecomposable division algebras, division algebras with nontrivial SK₁, ..., and connections with orderings on division algebras with involution. If time permits, we will also discuss connections between valued division algebras and graded division algebras.

Date received: April 29, 1999