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.