every ordered field with the least upper bound property is isomorphic to , proof that
Let be an ordered field with the least upper bound property. By the order properties of , and by an induction argument for any positive integer . Hence the characteristic of the field is zero, implying that there is an order-preserving embedding .
We would like to extend this map to an embedding of into . Let and let be the associated Dedekind cut. Since is nonempty and bounded above in , it follows that the set is nonempty and bounded above in . Applying the least upper bound property of , define a function by
We claim that in fact . To see this, first recall that since is a partially ordered group with the least upper bound property, has the Archimedean property (http://planetmath.org/DistributivityInPoGroups). So for any , there exists some positive integer such that . Hence the set is nonempty and bounded above, implying that lies in . Now observe that applying the least upper bound axiom in gives us that . Since is an upper bound of in , it follows that .
Seeking a contradiction, suppose . By the Archimedean property, there is some positive integer such that . Because , we obtain , which implies the contradiction . Therefore , and so . This completes the proof.
|Title||every ordered field with the least upper bound property is isomorphic to , proof that|
|Date of creation||2013-03-22 14:10:51|
|Last modified on||2013-03-22 14:10:51|
|Last modified by||mps (409)|