Decidable fragments of field theories

Journal of Symbolic Logic 55 (3):1007-1018 (1990)
Abstract
We say φ is an ∀∃ sentence if and only if φ is logically equivalent to a sentence of the form ∀ x∃ y ψ(x,y), where ψ(x,y) is a quantifier-free formula containing no variables except x and y. In this paper we show that there are algorithms to decide whether or not a given ∀∃ sentence is true in (1) an algebraic number field K, (2) a purely transcendental extension of an algebraic number field K, (3) every field with characteristic 0, (4) every algebraic number field, (5) every cyclic (abelian, radical) extension field over Q, and (6) every field
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DOI 10.2307/2274469
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References found in this work BETA
The Undecidability of Pure Transcendental Extensions of Real Fields.Raphael M. Robinson - 1964 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 10 (18):275-282.

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Algorithms for Sentences Over Integral Domains.Shih Ping Tung - 1990 - Annals of Pure and Applied Logic 47 (2):189-197.

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