Abstract We give necessary conditions for a set to be definable by a formula with a universal quantifier and an existential quantifier over algebraic integer rings or algebraic number fields. From these necessary conditions we obtain some undefinability results. For example, N is not definable by such a formula over Z. This extends a previous result of R. M. Robinson.

Let I be an algebraic integer ring. The decision problem of the solvability of diophantine equations with parameters in I is proved to be co-NP-complete. However, the decision problem of the solvability of diophantine equations with parameters in all algebraic integer rings is in P. Let $\varphi (x.y)$ be a quantifier-free arithmetical formula in conjunctive normal form or disjunctive normal form. The decision problem of the sentences of the form $\exists x\forall y\varphi (x,y)$, true in I, is NP-complete. Then the decision problem of the sentences of the form $\forall x\exists y\varphi (x,y)$, true in I, is co-NP-complete, whereas the decision problem of the sentences of the form $\forall x\exists y\varphi (x,y)$, true in all algebraic integer rings, is in P. Some other related decision problems are also proven to be in P.

Abstract We say φ is an ∀∃ sentence if and only if φ is logically equivalent to a sentence of the form ∀x∃yψ(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.

Abstract We study the sets definable in an algebraic number field by first order formulas of various simple types, showing in particular that N and Z do not have very simple definitions.