A construction of real closed fields

We introduce a new construction of real closed fields by using an elementary extension of an ordered field with an integer part satisfying . This method can be extend to a finite extension of an ordered field with an integer part satisfying . In general, a field obtained from our construction is either real closed or algebraically closed, so an analogy of Ostrowski's dichotomy holds. Moreover we investigate recursive saturation of an o-minimal extension of a real closed field by finitely many function symbols.