Some theories associated with algebraically closed fields

Abstract

We consider the effect on decidability of adding, to the decidable theory of algebraically closed fields of characteristic zero, relation symbols for algebraic independence or function symbols for differentiation. Our results show that the corresponding theories are usually undeeidable.Let k and K be algebraically closed fields of characteristic zero. Let K be an extension of k of transcendence degree n over k. Since k has characteristic 0, we may assume that the rational field, Q, is a subfield of k.Let Indn be the n-ary relation on K which holds for exactly those n-tuples from K which are algebraically independent over k.Let x1, …, xn be a transcendence base for K over k. For i = 1, 2, …, n, let Di: K → K be the partial differentiation function with respect to xi and this base.Let KnInd = (K, +, ·, Indn), n ≤ 1 and let KnDiff = (K, +, ·, D1, …, Dn), n ≤ 1 where K has transcendence degree n over k.We show that the theories of these structures are independent of k when k has infinite transcendence degree over Q, that KnDiff has undeeidable theory for n ≤ 1 and that KnInd has undeeidable theory for n ≤ 2. The theory of K1Ind is decidable.