Elementarily equivalent fields with inequivalent perfect closures

Abstract

We give a counterexample to the following conjecture due to L. V. den Dries: Let F , L F,L be two fields of characteristic p p . If F ≡ L F \equiv L then F 1 / p ∞ ≡ L 1 / p ∞ {F^1}/{p^\infty } \equiv {L^1}/{p^\infty } .