It is shown that the topology of
a topological field F is given by a complete, discrete valuation if and only
if F is locally strictly Iinearly compact. More generally, the topology of a
topological division ring K is given by a complete, discrete valuation and
K is finite dimensional over its center if and only if K is locally centrally
linearly compact, that is, if and only if K contains an open subring B, the
open left ideals of which form a fundamental system of neighborhoods of
zero, such that B, regarded as a module over its center, is strictly linearly
compact.
