Model-complete theories of pseudo-algebraically closed fields

Abstract

The model-complete, complete theories of pseudo-algebraically closed fields are characterized in this paper. For example, the theory of algebraically closed fields of a specified characteristic is a model-complete, complete theory of pseudo-algebraically closed fields. The characterization is based upon the algebraic properties of the theories' associated number fields and is the first step towards a classification of all the model-complete, complete theories of fields.

A field F ispseudo-algebraically closed if whenever I is a prime ideal in a polynomial ring F[x₁...x_m]=F[x] and F is algebraically closed in the quotient field of F[x]/l, then there is a homorphism from F[x]/l into F which is the identity on F. The field F can be pseudo-algebraically closed but not perfect; indeed, the non-perfect case is one of the interesting aspects of this paper. Heretofore, this concept has been considered only for a perfect field F, in which case it is equivalent to each nonvoid, absolutely irreducible F-variety's having an F-rational point. The perfect, pseudo-algebraically closed fields have been prominent in recent metamathematical investigations of fields [1, 2, 3, 11, 12, 13, 14, 15, 28]. Reference [14] in particular is the algebraic springboard for this paper.

A field F has bounded corank if F has only finitely many separable algebraic extensions of degree n over F for each integer n⩾2.

A field F will be called an B-field for an integral domain B if B is a sabring of F.