Ordered Exponential Fields
by
Salma Kuhlmann
Universität Heidelberg

Let K be a totally ordered field, and v be the natural valuation on K (i.e. the valuation induced by the archimedean equivalence relation). Note that the residue field [`K] is always archimedean, i.e. a subfield of the reals. Note also that v induces a topology on K, which coincides with the order topology in the nonarchimedean case. We say K is exponential if there is an order preserving isomorphism from the (ordered) additive group of the field onto its (ordered) multiplicative group of positive elements. We study criteria for a nonarchimedean ordered field K to be exponential. More precisely, if [`K] is an exponentially closed subfield of the reals, we give necessary and sufficient conditions on the value group v(K) so that one can extend the exponential on [`K] to an exponential on K. Moreover, we analyse in which cases the extended exponential inherits properties (e.g. the property of being differentiable with a derivative equal to itself, etc.) from the real exponential on [`K]. We also concentrate in general on the problem of extending an exponential function from a field to an extension field. In this context, extensions where the smallest field is dense in the larger one, immediate extensions, and power series extensions are treated with a particular emphasis.

Date received: April 12, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caae-51.