It is well-known that the field R of all real numbers is a realclosed
field and that, up to iscmorphism, it is the only Dedekind-complete
ordered field. Artin and Schreier generalized the algebraic properties of
the reals to form the rich, interesting theory or real-closed fields.
Among other things, they showed that any ordered field has an algebraic
extension that is real-closed, and which is uniquely determined up to
isomorphism. Many interesting non-Archimedean, real-closed fields F are
known. Under the interval topology, any ordered field is a topological
field. Under that topology, F is not Dedekind-complete, is not locally
connected, and is not locally compact.