A ug 1 99 7 The exponential rank of nonarchimedean exponential fields

Based on the work of Hahn, Baer, Ostrowski, Krull, Kaplansky and the Artin-Schreier theory, and stimulated by the paper [L] of S. Lang in 1953, the theory of real places and convex valuations has witnessed a remarkable development and has become a basic tool in the theory of ordered fields and real algebraic geometry. Surveys on this development can be found in [LAM] and [PC]. In this paper, we take a further step by adding an exponential function to the ordered field. Beforehand, let us sketch the basic facts about convex valuations. Let (K,<) be an ordered field and w a valuation of K, with valuation ring Rw , valuation ideal Iw , value group wK and residue field Kw. Then w is called compatible with the order if and only if it satisfies, for all x, y ∈ K: