On the structure of nonarchimedean exponential fields I

Abstract. Given an ordered field $K$, we compute the natural valuation and skeleton of the ordered multiplicative group $(K^{0},\cdot ,1,)$ in terms of those of the ordered additive group $(K,+,0,)$ . We use this computation to provide necessary and sufficient conditions on the value group $v(K)$ and residue field $\overline{K}$, for the $L_{\infty \omega}$ -equivalence of the above mentioned groups. We then apply the results to exponential fields, and describe $v(K)$ in that case. Finally, if $K$ is countable or a power series field, we derive necessary and sufficient conditions on $v(K)$ and $\overline{K}$ for $K$ to be exponential. In the countable case, we get a structure theorem for $v(K)$.