On~wild division algebras over fields of power series

A. B. Zheglov

Sbornik. Mathematics, Tome 195 (2004) no. 6, pp. 783-817

Résumé

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.