Stability: Index and Order in the Brauer Group

Abstract

A field is stable if for every division algebra $A$ in its Brauer group order of $A$ = index of $A$. Index and order in the Brauer group of a field $F$ with discrete valuation and perfect residue class field $K$ are calculated. Division algebras with specified order and index are constructed. For $F$ complete, necessary and sufficient conditions for the stability of $F$ are given in terms of the Brauer group of $K$. These results follow. A finite extension of a stable field need not be stable. The power series field $K((x))$ is stable for $K$ a local field. $K((x))$ and $K(x)$ are not stable for $K$ a global field.