Undecidability and Definability for the Theory of Global Fields

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.I.There is a finite collection of predicates which define every valuation, archimedean and nonarchimedean, of every global field, in terms of parameters.II.There is a sentence which distinguishes number fields from function fields. III. Given a global field K:If K is a number field, the theory of number fields defines its algebraic integers, the rational integers, and the natural numbers.If K is a function field, the theory of function fields defines its field of constants F, and for an arbitrary nonconstant x G K, defines the polynomial ring F[x] and a model of 'N given by the powers of x.Gdel functions encoding all finite sequences of elements of K exist for each of these models of N.This investigation was motivated by the papers of Julia Robinson [9], [10] and Ershov [4], who used the Hasse-Minkowski theorem on quadratic forms to prove the undecidability of a given number field, or field of rational functions over a finite field of odd characteristic.Its original goal was to show the remaining algebraic function fields were undecidable, completing the parallel between number fields and function fields.Then Simon Kochen observed that the proof showed