On the structure of nonarchimedean exponential fields I

Abstract

Given an ordered fieldK, we compute the natural valuation and skeleton of the ordered multiplicative group (K >0, ·, 1, <) in terms of those of the ordered additive group (K,+,0,<). We use this computation to provide necessary and sufficient conditions on the value groupv(K) and residue field\(\bar K\), for theL ∞ε-equivalence of the above mentioned groups. We then apply the results to exponential fields, and describev(K) in that case. Finally, ifK is countable or a power series field, we derive necessary and sufficient conditions onv(K) and\(\bar K\) forK to be exponential. In the countable case, we get a structure theorem forv(K).