On wild division algebras over fields of power series

Certain special classes of division algebras over the field of Laurent power series with arbitrary residue field are studied. We call algebras in these classes split and well-split algebras. These classes are shown to contain the group of tame division algebras. For the class of well-split division algebras we prove a decomposition theorem which is a generalization of the well-known decomposition theorems of Jacob and Wadsworth for tame division algebras. For both classes we introduce the notion of a $\delta$-map and develop the technique of $\delta$-maps for division algebras in these classes. Using this technique we prove decomposition theorems, reprove several old well-known results of Saltman, and prove Artin's conjecture on the period and index in the local case: the exponent of a division algebra $A$ over a $C_2$-field $F$ is equal to the index of $A$ if $F=F_1((t))$, where $F_1$ is a $C_1$-field. In addition we obtain several results on split division algebras, which, we hope, will help in further research of wild division algebras.