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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