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Finite Codimension Subfields of a Field Complete with Respect to a Real Valuation

ABSTRACT Let K be a field complete with respect to a real valuation v and not algebraically closed. We will show that every finite codimension subfield of K is closed in the v-adic topology if and only if the degree of imperfection of K is finite. It follows that there are incomplete finite codimension subfields of K when the degree of imperfection of K is infinite. These examples exhibit other interesting pathologies. We are able to give a necessary (and in the case of a discrete real valuation also sufficient) condition for a given finite codimension subfield to be complete. Finally, we give some applications to fields of Laurent series. Communicated by A. Prestel.