Pseudofinite fields, procyclic fields and model-completion

Abstract

The first two sections of this paper contain some basic definitionsand results necessary for an understanding of the main theorems. Theseare stated at the end of §1. As is well-known, a model completion of atheory is not determined by the models of the theory alone, but also bythe language in which the theory is formulated. This makes the choiceof a model-completion rather arbitrary. The advantage of the particularlanguage adopted here for the theory of procyclic fields is that not onlydoes one obtain a model-completion of the theory (Theorem I) but onecan recover the models of the theory as the class of all substructures ofthe models of the model-completion for which the defining axioms of theextended language hold (Theorem 2). This is proved using the results ofAx [1, 2] and Jarden [3]. It is worth remarking that in this language, thetheory of pseudofinite fields has elimination of quantifiers [Kiefe,4]. Also, the authors wish to acknowledge that the key idea in the proofof Theorem 2 was inspired by A. Robinson [6].1. Let r be a similarity type, L