On the Galois theory of division rings

Abstract

1. Throughout this paper, K will represent a division ring and L a galois division subring. We are interested in establishing a galois theory for the extension K/L when K/L is locally finite. In order to do this one must identify the galois subrings of K containing L. An example given by Jacobson [4] shows that not every such division subring is galois. However, we obtain that each subring subject to a natural finiteness assumption is galois when the dimension [K: H], <?o where H is the division subring left fixed by all inner automorphisms which leaves L fixed. For these subrings we then obtain the usual theorems. We rely strongly on topological characterizations of the galois groups. In particular, we characterize the extensions with locally compact and complete galois groups. These results provide extensions and simplications of the recent results of Nagahara, Tominaga, and Nobusawa on division rings cited in the bibliography.