In this part we develop the theory of finitely determinate structures, that is, structures on which the dual quantifiers “stat” and “unreadable” have the same meaning. Among other general
In this part we develop the theory of finitely determinate structures, that is, structures on which the dual quantifiers “stat” and “unreadable” have the same meaning. Among other general
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...xm]=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.
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