Amalgamation and Elimination of Quantifiers for Theories of Fields1

The universal theories of integral domains and of ordered integral domains which have the amalgamation property are characterized via their existentially complete models. The results of A. Macintyre, K. McKenna, and L. van den Dries on fields and ordered fields whose complete theories permit elimination of quantifiers are then derived as easy corollaries. A. Macintyre, K. McKenna, and L. van den Dries [5] have shown that the only theories of fields which permit elimination of quantifiers (in their natural languages) are those which previously had been known to do so. Namely, the only infinite fields whose theories permit elimination of quantifiers are the algebraically closed fields. The only ordered fields whose theories permit elimination of quantifiers are the real closed ordered fields. Analogous statements hold for valued fields, formally p-adic fields, and formally w-adic fields. These assertions are converses to well-known results of A. Tarski and A. Robinson and more recent results of A. Macintyre. The purpose of this paper is to show that these converses follow from more general results concerning which universal theories of integral domains have the amalgamation property. For the case of ordinary fields, the crux of the matter is that if a universal theory T of integral domains has the amalgamation property, then the infinite, existentially complete models of T are algebraically closed. The result on elimination of quantifiers then follows from the connection between amalgamation and elimination of quantifiers and from standard facts about model-companions. §1 contains preliminary information and the connection between amalgamation and elimination of quantifiers. §2 considers the case of fields and integral domains. §3 considers the case of ordered fields and ordered integral domains. §4 contains comments on the other cases and also raises a question for further investigation. Not surprisingly the proofs in §§2 and 3 use some of the techniques developed in [5]. Received by the editors July 31, 1978. AMS (MOS) subject classifications (1970). Primary 02H05, 02H15, 02G20; Secondary 12L99.