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 Mat. Sb., 2004, Volume 195, Number 6, Pages 21–56 (Mi msb825)

On wild division algebras over fields of power series

A. B. Zheglov

Humboldt University

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.

DOI: https://doi.org/10.4213/sm825

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English version:
Sbornik: Mathematics, 2004, 195:6, 783–817

Bibliographic databases:

UDC: 512.552.32
MSC: Primary 12E15; Secondary 12J10, 16K20, 16K40, 16K50

Citation: A. B. Zheglov, “On wild division algebras over fields of power series”, Mat. Sb., 195:6 (2004), 21–56; Sb. Math., 195:6 (2004), 783–817

Citation in format AMSBIB
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