On Exponentially Closed Fields1

It is well known [4] that the non-Archimedean residue class fields K of the ring of continuous real valued functions on a space are realclosed and 7/i-sets. It does not appear to be known that the exponential function in the reals induces an exponential function in K (definitions to follow) ; thus K is exponentially closed. The property of being exponentially closed is a new invariant which will be applied to totally ordered fields in this paper. A totally ordered field K will be called exponentially closed if (i) there exists an order preserving isomorphism / of the additive group of K onto K+, the multiplicative group of positive elements of K, and (ii) there exists a positive integer « such that l + l/«</(l) <«; such an isomorphism will be called an exponential function in K. In §0 Archimedean exponentially closed fields will be considered, the rest of the paper being devoted to the non-Archimedean case. In §1 some necessary conditions for a non-Archimedean field to be exponentially closed will be given, followed in §2 by some examples. In §3 a set of sufficient conditions will be given, followed by an example. A totally ordered field K will be called root-closed if K¥ is divisible. Clearly exponentially closed fields and real-closed fields are rootclosed.