Abstract:
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.