Let F be an ordered eld v the unique nest valuation on F compatible with the ordering so v a v b i njbj jaj for some integer n V the value group of v and the residue eld of v so is archimedean If F is an ordered extension of F then the nest valuation on F compatible with the extended ordering is an extension of v which we denote also by v We denote by V and the value group and residue eld of the extended v F denotes the real closure of F The residue eld in this case is the real closure of The value group is V the divisible hull of V Any power series eld is maximally complete V p p V is algebraically closed so V is real closed The natural valuation on V denoted also by v is the unique nest valuation on V compatible with the ordering The following is a summary of our results Suppose a proper embedding see Section for the de nition p F V is given and F is an ordered extension of F generated by a single element y We show there exists a canonically de ned power series V such that p extends to a proper embedding p F V via y Using results of Mourgues and Ressayre we show that p F truncation closed p F truncation closed We also show that p F e V e V e denotes the smallest subextension e closed under adjoining n th roots of positive elements n Of course if R then e Roughly this is what is asked for by MacLane and Schilling in Final Remark It allows us to read o the extended value group V directly from the power series

In the paper it is shown how an embedding of an ordered field F into a formal power series field can be extended canonically to an embedding of any simple extension F(y) of F. Properties of the extended embedding are studied in detail. Several applications are given. (C) 2002 Elsevier Science B.V. All rights reserved.