It is well known that if D is a
Dedekind domain with quotient field K and if T is any Hausdorff nondiscrete field
topology on K for which the open D-submodules of K form a fundamental system of
neighborhoods of zero, then T is the supremum of a family of p-adic topologies. We
show that if the class number of K over D is finite and if T is any Hausdorff
nondiscrete field topology on K for which D is a bounded set, then T is the
supremum of a family of p-adic topologies. We then investigate the problem of
extending a locally bounded topology from D to a locally bounded topology on K.
The extendable topologies on D for which there exists a nonzero topological
nilpotent and for which D is a bounded set are characterized. Moreover
it is shown that the topology of a locally compact principal ideal domain
A extends to a ring topology on the quotient field of A if and only if A is
compact.
