Geometric characterizations of existentially closed fields with operators

Pierce D.

ILLINOIS JOURNAL OF MATHEMATICS, vol.48, no.4, pp.1321-1343, 2004 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 48 Issue: 4
  • Publication Date: 2004
  • Doi Number: 10.1215/ijm/1258138514
  • Journal Name: ILLINOIS JOURNAL OF MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.1321-1343
  • Middle East Technical University Affiliated: No

Abstract

This paper concerns the basic model-theory of fields of arbitrary characteristic with operators. Simplified geometric axioms are given for the model-companion of the theory of fields with a derivation. These axioms generalize to the case of several commuting derivations. Let a D-field be a field with a derivation or a difference-operator, called D. The theory of D-fields is companionable. The existentially closed D-fields can be characterized geometrically without distinguishing the two cases in which D can fall. The class of existentially closed fields with a derivation and a difference-operator is elementary only in characteristic 0.