Nontrivial order preserving automorphisms of non-Archimedean fields

A study of order preserving field automorphisms of ordered nonArchimedean field extensions of R will be presented. We show that, while the identity map is the only field automorphism of R, infinitely many nontrivial order preserving field automorphisms can be constructed on an ordered nonArchimedean field extension F of R. Moreover, we show that if P is an order preserving field automorphism of F then P (r) ≈ r for all r = 0 in R ⊂ F ; and invoking the axiom of choice, we construct an order preserving field automorphism P of F satisfying P (r) = r for infinitely many r ∈ R.