We say \varphi is an \forall\exists sentence if and only if \varphi is logically equivalent to a sentence of the form \forall x\exists y \psi(x,y), where \psi(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 \forall\exists 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 \mathbf{Q}, and (6) every field.