Wild division algebras over Laurent series fields

Abstract

In this paper we study some special classes of division algebras over a Laurent series field with arbitrary residue field. We call the algebras from these classes as splittable and good splittable division algebras. It is shown that these classes contain the group of tame division algebras. For the class of good division algebras a decomposition theorem is given. This theorem is a generalization of the decomposition theorems for tame division algebras given by Jacob and Wadsworth. For both clases we introduce a notion of a $δ$-map and develop a technique of $δ$-maps for division algebras from these classes. Using this technique we reprove several old well known results of Saltman and get the positive answer on the period-index conjecture of M.Artin: the exponent of $A$ is equal to its index for any division algebra $A$ over a $C_2$-field $F$, when $F\eq F_1((t_2))$, where $F_1$ is a $C_1$-field. The paper includes also some other results about splittable division algebras, which, we hope, will be useful for the further investigation of wild division algebras.