The model-complete, complete theories of pseudo-algebraically closed field:; arc characterized in tilis paper, For example, the theory of algebraically closca lields of a specified characteristic is a model-complete, complete theory of pseudo-algebraically closed fields. The characterization is based upon algebraic properties of the theories' associated number fields and is tile first step towards a classification {,fati the model-complete, complete theories of fields.
A field 1:" i~ pseudo-algebraieally closed if whenever I is a prime ideal in a polynomial ring F[x, ..... x,,.] = F[x] and F is algebraically closed in the quotient field of F[x]/I, then there is a homorphism from F[x]/I into F which is the identity on F. The i~eld 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 equi,alent to each nonvoid, absolutely irreducible F-variety's ha~ing an F-rational point. The perfect, pseudo-algebraically closed fields ha,, e been prominent in recent metamathematical investigations of fields [l, 2, 3, 11. 12, 13. 14. 15, 28]. Refclenee [14] in particular is the algebraic springboard for this paper.
A field F has bounded corank if F has on b finitely many separable algebraic extcnslons of degree n over F for each integer n > 2.
A field F will bc called an B-field for an integral domain /3 if B is a s.~bring of F.
Some of rite results of this paper are the following:
Theorem. A complete, consistent theory of psemlo-algebraically closed B-fiehts is model-enmplete if and only if its associated B-number fieht tl has bounded corank, in which casc the gicen ~hΒ’ory is the model-eon:panion of a coulpnnent of the theory T~ a]' Iotally transcendental extensions of E.