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).
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This paper represents some results of the author's doctoral thesis
This paper was written while the author was supported by a research grant from the University of Heidelberg
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Kuhlmann, S. On the structure of nonarchimedean exponential fields I. Arch Math Logic 34, 145–182 (1995). https://doi.org/10.1007/BF01375519
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DOI: https://doi.org/10.1007/BF01375519