Publisher Summary This chapter describes locally bounded topologies on the rational field Q. Their description yields a new characterization of the classical division rings, R, C, and H, and a characterization as topological rings of real commutative Banach algebras with identity. Basic information about Dedekind domains is presented. Classification is given of those ring topologies on the quotient field of a Dedekind domain D for which the open D-submodules form a fundamental system of neighborhoods of zero (in particular, of those topologies on Q for which the open additive subgroups form a fundamental system of neighborhoods of zero). To describe locally bounded topologies on the rationals and related fields, one need a new method of constructing a topological ring from a family of topological rings, each having a designated open subring.