Abstract.We prove that the theory of global fields is essentially undecidable, using predicates based on Hasse's Norm Theorem to define valuations. Polynomial rings or the natural numbers are uniformly defined in all global fields, as well as Godel functions encoding finite sequences of elements.We will prove that the elementary theory of global fields is essentially undecidable. While this is a negative logical result, our proofs have the positive consequence that a great variety of number-theoretic objects, from rings of integers and valuations, to zeta-functions and adele rings, can be discussed in the theory of global fields. It is our hope that the theory may eventually be a vehicle for applying logical methods in number theory.Our main theorems are as follows.